Explain what is a clean power amplifier?

With regards to power line noise, on a separate front from distortion, that is not the amp but the electricity that runs it, and a truly clean amp probably has a good quality power supply that does what is needed to feed the amp clean power it needs to perform well. If not external power conditioning should help or be an insurance policy at minimum.

True that both noise and distortion makes an amp "unclean".  The amp itself produces both to various degrees aside from the electricity it runs on. 

The horror!   What a mess!

Clean is a subjective term. 
What is a “clean” house?   No house is perfectly clean.  It’s just a matter of how much dirt one tolerates.  
With amps it’s distortion, not dirt.   All amps have some.   Can you hear it?   If not it sounds clean.   
Also all amps produce more distortion when they have to work harder so as the distortion goes up they become less “clean” sounding. 
At some point they all run out of juice and clip.  That’s when things really start to become a mess. 
Hope that helps.