Do you get that "breath noise" when playing single notes or only when playing chords? Does it depend on how loud you have the amp?
I at one time played quite a bit through a Classic 20 and it is an OK harp amp (although not very loud), but needs a lot of "fiddling" with the settings before you understand how to set it.
I'm guessing that it isn't "breath noise" at all but distortion that occurs when you have the gain on the pre-amp turned up too much. This amp has "master" and "volume" controls, one of which is the pre-amp gain and one of which is the power-amp gain. (I don't remember which is which...) You want the pre-amp set pretty low, as pre-amp distortion sounds fairly ugly and if you have the gain high here it tends to feedback easily. You want power tube distortion instead - as I understand it, pre-amp distortion gets re-amplified in the power tubes and can turn into "noise", while power tube distortion goes straight to the speaker and gets turned into gritty music..... It can help to swap the pre-amp tube - stock is 12AX7, using a 12AY7 or 12AU7 (lower gain tubes, I think) makes the amp sound better and have more usable range.
I could be wrong - maybe it is breath noise - but I remember having the sound "break up" a LOT when the settings weren't right!