What's more important in a difficult room, room correction or higher, clean, power?

If you use a distributed bass array such as the Swarm, standing waves are eliminated and get get much more even bass throughout the room, not just at the listening chair.

Years ago I did a Stereophile show in NYC. We had room correction and also a standing wave that was killing the bass at the listening chair. Now to be clear all room correction does is generate an EQ curve such that you get flat response at the microphone location. So if there is a dip in the bass as might be seen with a standing wave in the room, the room correction will simply demand more power of the amps at the bass frequency dip. It really doesn't matter how much power you put into it; if a standing wave is present the dip **will** persist. Once I realized that I also realized that our room correction (Accuphase) was simply causing coloration at other frequencies; as soon as we removed it the system sounded more transparent and neutral. In the end we simply lived with the standing wave.

That was before DBAs (Distributed Bass Arrays) were available. Up until I heard what the Swarm (which is the first application of this principle that I know of) actually did I really had no time for subs as it was so difficult to get them to blend since the bass was different depending on where you were in the room. The Swarm DBA system won me over to subs.

Sorry, I meant to say the TacT computes at 96 kHz.

1 - Room treatment2 - EQ

In that order.

So, @mijostyn, what unit are you using that does all of that?  I would have guessed a TacT, except those compute at 24 kHz.

What's important (as others have said more verbosely) is to fix the acoustics. It cannot be done with more power.

IMO it is best done by First, positioning listener, main speakers, and subs for smoothest response; Second, adding bass trapping and 1st-reflection treatment if necessary; and Third, using DSP if you want to smooth really low frequencies, say below 80 Hz, further.

In doing all of that, measurement capability (e.g., REW and a calibrated mic) helps a lot.

1+ Ralph. As Ralph says no amount of digital correction will fix a standing wave. For most of us you have to use multiple subwoofers. This does not mean that room control does not have a place. The best room control will provide perfectly flat frequency response across the entire audio spectrum and assure that the response from both channels is exactly the same. This gives the best image. From there you can modify the frequency response to taste and do some very cool stuff. My unit has dynamic loudness control. It automatically jumps from one Fletcher Munson curve to the next with changes in volume. It is programmed with 8 Fletcher Munson curves which can be modified to taste. The end result is that the spectral balance remains the same from -40 dB to 0 dB. Bas management is insane. You have independent control over high pass and low pass filters at 1 Hz increments from 0 to 375 Hz with slopes from 1st to 10th order and you can change it all on the fly. The unit will keep nine frequency response curves in memory which you can switch by remote control. My normal curve slopes down from 1 kHz to 6 dB down at 20 kHz. I have another curve down 12 dB at 20 kHz for bright records and another with a 3 kHz notch filter for sibilant records. There is no way you can do these things in the analog world and for those of you worried about resolution this is all done 192 kHz 48 bit.    

The room modes below 50Hz were best avoided by careful speaker and listening position placement in my room. Corner traps helped somewhat along with panels.

Dirac Live with the filter on still sounds markedly better. Mixed phase filters with phase time alignment and magnitude correction creates vivid, lifelike soundstages. The usb signal exits the PC digitally corrected, so no additional A/D anywhere. Agree that deep nulls caused by room modes are not solved by room correction nor are first refllection points, but what about everywhere else? I'm so happy I can flick the Dirac filter switch to on and let my ears decide.

Room correction was the most important factor in my tweaks, and easily the most dramatic improvement of my audio system ( I am in a small room and use reflective and absorbing surface in some ratio+ cheap schumann resonators+ many Helmoltz bottles homemade) ...But before that it is very important to heavily controls the vibration resonance problem linked to all the gear and speakers....After that room treatment will be possible and astounding if you make it happen with your ears open...In my case the cost was low because I only use homemade or cheap products, and even with that the change was epic toward holographic sound and imaging and natural timbre of instruments...

I have had room correction based subwoofers and they are limited as to what they can correct in the bass response.  
They definitely do not satisfactorily correct issues that are caused by poor fundamental integration between room and main speakers.  
Fundamentals include best position in room for main speakers, best position for subwoofer, proper subwoofer crossover frequency and phase and proper subwoofer level.  
Once these were set well the sound was excellent and I could not tell any difference when room correction was engaged.  
My recommendation would be to get some RTA software (phone, laptop or tablet) and a CD with full bandwidth pink noise and start measuring the response and then adjust positions, crossover, phase etc until you have a smooth response.  It is always comforting to know your response is smooth.    Then "season" to taste, slight adjustments in level to suit your preferences. 
  

Re the OP’s scanario:

"Tryin’ to get high, without having to pay..." -Marianne Faithfull

Room correction is not a fix, the effect of the room problems still exist and would still be interfering. It’s like a continual hasty pasting over of an ongoing audio murder.

Room correction can be done judiciously, minimally...and then one has to figure out if the given minimal corrections are worth the problem of interference in signal purity that is brought about via the installation and use of the hardware the creates the corrections.

Then to understand such under the given signal disturbances and coloration that is carried out by the choices in the rest of the gear that is in the room.

physical acoustic correction via physically correcting the room is the best bet, every time and likely always will be. It is just the nature of how acoustics and audio works... that makes active correction suitable for non fidelity situations like concerts, malls, airports, bars, commercial theaters, etc...places where fidelity is not the primary concern.

If DSP was the fix that it is advertised to be by some, then it would have taken over the high fidelity world by storm.

Even though it has been around for quite some time, note that it has not taken over.

Thus, thinking cap..a bit of funds...some work..and fix acoustics.

IF this cannot be done, then the room must be accepted as it is, or try some minimal DSP correction, with the caveat that the carrier of the DSP, the digital systems involved - are destructive to overall fidelity.

One can fix some of the annoying bits with the DSP and then find themselves listening less and less to music in that space, as the positives are outweighed by the negatives of the digital manipulation. Negatives that take time to consciously discern and put label to.

In my 12 * 22 * 7 foot room, with an additional space under the basement stairs that makes the room L shaped, I got pretty good bass definition with my Vandersteen 5A Carbons. But there were some bad dips between 72 and 120 hz that I could not sufficiently smooth out, up to at least 10 dB below 70, the middle point on my meter. So I purchased three GIK Acoustics Soffit bass traps that I placed at the two corners on either side of the speakers, arranged on the long wall, and one directly behind my chair, in the nook under the basement stairs, where there was a lot of bass buildup. A few weeks later when I had some free time, I readjusted the bass on the speakers. The results showed some smoothing in the measured frequencies, but there were still some dips in that 72 to 120 hz range that I could not correct, which remained down 8 to 10 dB. However, I was not prepared for how dramatic the improvement was in terms of slam. The lower bass is dramatically more powerful and cleaner. Even with the equalized bass, that was not good enough to achieve what I’m hearing now. So, I can definitely vouch for room treatment! Aesthetically speaking, I’m done. I could probably get even better results with more traps, but I don’t want any more traps in my room.

The problem with room treatment is the amount of it you need to treat low frequencies, that's where I think DSP can be useful.

This is also where a distributed bass array like the Swarm is helpful.

I usually think of room treatment not room correction. It might be of help on the sub, if not don't use the correction feature. If you like the Peachtree you should keep it. 

I'm not a fan of room correction. The reason is most often what is going on is a standing wave which no amount of correction will fix.

You can kill a standing wave by using a distributed bass array. The best example of this is the Swarm made by Audiokinesis. The Swarm is four subs designed to sit right against the walls. You place them asymmetrically about the room; in this way they produce consistent bass across the spectrum regardless of where you are in the room.

I should have clarified my reference to 'power' in that I'm not looking for cleaner AC power but whether my 150 WPC Nova 150, for example, is a better amp/integrated solution than say an amp/integrated that is rated at a lower WPC but has room correction built-in.

Yes, you should have been a lot more clear.

Now the answer is easy. Anything with room correction, don't even consider it. 

The one thing you are onto is that a lower powered amp might well be better. The guiding principle was best stated by the great Robert Harley many years ago, "If the first watt isn't any good, why would you want 200 more of them?"

Have a read of this... might be worth thinking about different speakers. DSP on a sub is never a bad idea though.

I should have clarified my reference to 'power' in that I'm not looking for cleaner AC power but whether my 150 WPC Nova 150, for example, is a better amp/integrated solution than say an amp/integrated that is rated at a lower WPC but has room correction built-in.

As I noted, I'm considering an Elac EA101EQ-G integrated amp that has room correction but it has a significantly lower WPC rating than my Nova 150.

My Elac Debut B6.2 speakers like power, as did my previous KEF Q100's, but I don't think either fully delivers what they're capable of due to my difficult room.

You don't need to take the plunge, and you don't need to ask other people. This is something you are perfectly capable of figuring out for yourself and all by yourself, and you already have everything you need to do it.

Plop your sub down somewhere, anywhere, play some music with regular steady low bass, move slowly around the room, and listen. Note the places its really loud, and note the places you can hardly hear it. 

Now move the sub and repeat. Notice some of the places it was too loud are now too quiet. Notice this changes depending BOTH on where you stand to listen, AND where you place the sub.

Now ask yourself, if I use room correction, which location am I going to correct for? Really think about it. Eventually you will realize room correction can make it sound fine only in the one spot. Everywhere else it makes worse.

If that is what you're after, and you like spending money on stuff like that, then go for it.

Now for your next lab project, what will cleaner power sound like? For this one first get your system all nice and warmed up. Sit yourself down in the sweet spot and play your favorite recording. Play the whole thing. Relax. Enjoy.

Now go out to your breaker panel, flip off all the breakers, leaving on only the ones you need for the system, maybe a few lights. Everything else you flip off.

Go back in and listen to your favorite track again. Or any track, for that matter. I'll be surprised if you're not shocked how much better it sounds. Everyone I've ever done this for sure has been.

The improvement you heard, you can get at least that much from a good power conditioner, power cord, stuff like that. Note I said "good". Most of what's out there is not that good. But now, in addition to discovering you don't need room correction but do want cleaner power, you also now have a baseline reference to judge your cleaner power by.

This is what we used to call teach a man to fish. As opposed to give the man a fish. Now which, I wonder, will you choose?