I have yet to see a solid state device of any kind that did not benefit from at least 24 hours of warmup.
Certainly in this case its worth a try- if it makes no difference then don't leave it on all the time :) But I think you will find that it does make a difference.
Now- will that fix the mushiness at high volume? Hard to say. If it does not, its probably not because of insufficient power; if a speaker needs more than this amp makes then the speaker is simply impractical for the real world. But if there is not enough filter capacitance in the power supply (or if the filter caps in it are having some issues), mushiness at high volume will be the result, regardless of the power the amp makes or how long it is left on.