I research the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800.
Note 1 - Without a KNOWN county, the chance of an ancestor's rcds NOT being in this database is AT LEAST 35%.
Note 2 - Only 1/2 of Co. Cork's rcds are in this database.
Note 3 - The RC church kept control over most parish rcds from Tipperary (north). These are ONLY available thru the Tipperary Family History Research center at a substantial fee (regardless of outcome).
For James Cahill B 1838 +/- 4 yrs in Annaghdown, Co. Galway, with a father forenamed Thomas, I found 1 rcd in 1838. This is highly likely to be for your ancestor.
For Thomas Cahill M before 1839 ditto, I found 0 rcds.
For Bri(dget) Feeney M ditto, I found 2 rcds: 1836 and 1838. However, without a matching rcd for Thomas, these are likely false positives.
For Bri(dget) Feeny B 1817 +/- 4 yrs in Co. Galway with the father's forename unknown, I found 0 rcds.
Google the Foundation's name to lead you to this database. Register, opt to be notified as rcds are added and browse. Pls post your search results to my genforum msg.