I've found just a very few references to the song as Come Back Barney.

Edith Fowke recorded a set in Newfoundland in 1960, but I don't think it's been published; there also seems to be a set noted by Lucile Morris Upton at The Western Historical Manuscript Collection, Columbia; but that is a catalogue reference only.

You can see a text noted by Nathan D. Rose from Seamus Duffy, Aghamore, County Mayo (1988) at http://villasubrosa.com/Nathan/texts/duffytext.html#4

On the face of it, it looks to be American "Stage Irish", but there aren't many references available at the moment; so that's only a guess.