mooch "to dawdle in a bored or listless manner" is from mooch "to loaf, to skulk, to sneak". This is probably from Anglo-Norman muscher "to hide", as in Middle French muce, musse, mouce "hiding place". This is borrowed from an unattested Celtic form that is from the same base as Old Irish múch "smoke", Welsh mwg "smoke" (OED). Welsh mwg is from Proto-Indo-European *(s)meug- "to smoke". This became Old English smocian, English smoke.
No relation to French mouchoir or mouche.