Again, it depends. I depends on the actual noise/distortion of the AC mains power and the design of the power supply.
Below is a simple example.
View attachment 1267872
This can apply to the power transformer of our vintage gear. Once the common mode noise, and note that it is
common mode noise, makes it past the power transformer, it can or may
modulate (change), for want of a better term, the audio signal. This may or may not be subjectively audible to the listener.
I am not sure that I follow your question. The basic component of our AC mains power will be 50HZ/60HZ.
Of course you can take a 1 GHz sine wave and add to it the first 2 odd harmonics and get the same waveform shape as doing the same thing with a 50HZ/60HZ sine wave.
For those that say the internal parts of the power supply are more than adequate, see the picture below, and yes it is somewhat of an exaggeration, just used for the sake of clarity.
View attachment 1267893
Okay for the sake of example, lets say that the beginning of a powerful musical note happens at the exact time as the dip (the red circles) in the AC mains waveform. What do you think will happen?. The power supply will be starved for voltage. Or what do you think will happen if the power supply is repeatedly, periodically starved for voltage. This may not do nice things to the audio output.
Again, my example is somewhat of an exaggeration for the sake of clarity. But even a smaller amount of distortion may result in a change in what is called the load step response of the power supply. Depending on its severity, it may or may not make a subjective difference in the sound quality.
Again, see my post regarding common mode noise. Not all power supplies are created equal when it comes to dealing with common mode noise.
To be clear, I am not saying that outboard AC mains power conditioning is always needed for everyone and that there will always be a subjective change in the sound quality.