I found reference to a song called "Sweet Ireland Mary" here, on PDF page 7 (actual page 14). It was referred to as a Morris Dance tune by Charles "Cocky" Turner, who learned it from his dad Charles Tanner, who danced starting in the 1830s. Unfortunately, the source for the citation is cut off of the pdf, so you'd have to track down an original copy, or if it's any different from The Irish Girl, since it does show up in that section of the song.

Otherwise, sadly, I'm not finding much -- all the words and phrases are too common