I searched 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. Unless the county is known, there is AT LEAST a 35% chance the ancestor's rcds aren't in the database.
For Andrew Mooney B 1800-1810, I found 1 B rcd: 1807, Co. Meath. His father's name was Pat(rick).
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.