Interesting that the Bodleian Libray 1850 source quotes "Musha ring a ding a ra". this is mush closer to some recognisable Irish.
As has been said, "musha" is probably "maise" (indeed/sure/certainly etc))
"ring a" could be "rince" (dance/dancing, pronounced "rinka") and "a ra" is "a rá" to say or be said as in the sentence "tá rúd eigin aige a rá" meaning "he has something to say", (literally "is thing some at him at a saying") .
Doesn't exactly translate the chorus I iknow, but at least we can see that it probably did start life having something to do with music, dance etc!
Slán! Dáithí