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 Ann(ie) Hawkes B ???? somewhere in Ireland with the father's forename unknown, the earlies B rcd I found is in 1822 Kilkenny. Too late for your timeline.
For Henry McCloskey B ditto, the earliest rcd I found was in 1799 Louth, the next is in 1822 Tyrone. Neither fits your timeline.
Your McCloskey origins must have been protestant and lived in northern Ireland. Rcds for this sect and from this area are always iffy. There is a small chance that they were from Derry. Derry rcds haven't been transcribed yet.
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.