If I turn off the amp while it is working and wait few mins and turn it back on, the sound comes instantaneously. It’s only that if the entire system is off (for few hours) and I just turn it on, the sound doesn’t come for 1-3 mins until I hear a click sound from the amp?
The click is a relay. The amp is waiting until the power supply is fully charged and stable. Maybe a few other things too. Different amps have different circuits for this. My Herron phono stage takes a good 1 to 3 minutes same as this. My tube integrated has no relay but when first turned on after a long time takes a good 30 seconds to a minute during which if you're playing music it sounds like someone started with the volume control at zero, waited, and then started turning it up. Takes about a minute getting to full volume. So what you have sounds perfectly normal to me.