Irish Molly is an American song written by William Jerome, with music by Jean Schwartz; you can see two examples of 1905 sheet music at the Lester Levy Sheet Music Collection:

My Irish Molly O
My Irish Molly O

It's a typical product of the American "Stage Irish" style; such songs very often include tags such as "acushla" and "machree", which essentially were thrown in for flavour by professional songwriters who as often as not were far from being Irish themselves. Although they will have known that the terms were common endearments, they may well have had no idea of their derivation or precise meaning. Those two terms in particular have, I think, been discussed here a number of times; they are very common in stage and parlour songs of the period.