I searched the Irish Family History Foundation's database. It contains 11,500,000 surviving rcds for abt 60% of Irish counties. Rcds B4 1840 are sparse. Only 1/2 of Cork's rcds are online.
For Patrick Fallon, B 1861 +/- 4 yrs, I found 65 B rcds.None showed a middle name of Clayton. This is not unusual. 14 are in Galway. For a DOB of 1861 (exact), I found 7 B rcds. One is in Galway. This is a more manageable number provided your DOB is correct.
For Thomas Fallon B 1874 +/- 4 yrs, I found 61 B rcds. For 1874 (exact), I found 4 B rcds, but none in Galway or Cork.
For Nora, there is no way to search without the maiden name. Suggest you look at Kentucky M rcds to find it. Other sources are census rcds, obits and naturalization papers. Also, Nora may have lived long enough to have been eligible for Social Security. SS has a death index. This is at SSDI, BUT you need to know her date of death. Birth rcds for Patrick and Nora's children may also give her maiden name.
You can assume that Patrick's father was named John as a working hypothesis.
I hope this helps. Google the Foundation's name to lead you to this database. Register, opt to the notified as rcds are added, and browse. Pls post your search results to my Genforum msg.