I'll have a go.
Let's assume that any circuit has a certain amount of power supply noise rejection ratio. The higher this PSRR is (assuming equal bandwidth) the more immune to noise on the incoming PSU voltage rails it will be. So a great amp with great PSRR will be immune to all but the craziest of mains variances, it doesn't matter if you give it a wall wart of a massive over specced power supply, as long as it can fill its psu capacitors it's happy because the circuit after the psu is so good it can deal with anything without audible effects, my Modulus 86p is one such amp is has over 90db of PSRR, you could feed it with a square wave and it wouldn't care. The opposite end of this spectrum is something that has woeful PSRR, like a Naim 72 gain card which only has a cap and a resistor as the power rail filter. So my mod 86p doesn't benefit from a better psu, but Naim pre-amps (of a certain age) do.
Obviously it's way more complicated than just that as there are myriad ways the noise can manifest in the output signal, but basically there's circuits with high PSRR over a wide bandwidth and then there's the other stuff.
Now some people try to make the argument that only a better amp can resolve the audible benefits that come from a better PSU, but honestly that's a logical fallacy of the first water. If the measured performance, (sound) of a circuit is easily affected by changes the quality of the power supply entering the box then the circuit has poor PSRR and can be improved. Some people say " ah but does it sound good just because it has high PSRR", well if it has low measured distortion and load invariance, and all you really want is fidelity to the source, then yes, it should/must sound good. If low distortion doesn't sound good to you then you're after an effects box not a hifi.
In truth, most people want a bit of an effects box, even if they won't admit/accept it. (Only Serge is immune)..