I have not been able to determine a middle name for Mary Irish. The cemetery index for the South Dryden Cemetery in Lapeer Co, MI. lists her simply as Mary Irish Stafford. Another researcher visited that cemetery and found the her headstone is in bad shape (weathered and in pieces with some scattered) and cannot be read at this time. Perhaps the pieces can be found and reassembled at some point.The death record of a daughter, Cordelia Ferguson (d. 1904) names her mother as simply Mary Irish with no middle name.
Of course the mention in Bates of "Polly" and "May" is ambiguous. I feel that one of those girls must be Mary Irish as Deborah Irish is almost certainly the wife of Pardon, Deborah Soper. Who the other girl would be I have no idea (assuming that the reference in Bates is correct).
Sorry that I cannot add more information at this time. If I come up with something furher I will be sure to post it here.
I wanted to ask... What is ancestry of the Reuben Irish that you cite? How would he relate to the Irishes living in Erie County, PA circa 1820?