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 parish, 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 William Monahan B 1817 +/- 3 yrs somewhere in Ireland with the father's forename William, I found 4 rcds: 2-1813 Meath, 1814 Meath, and 1817 Down.
For William Monahan M 1843 +/- 1 yr somewhere in Ireland, I found 3 rcds: 2-Meath, 1-Longford. One of these MAY be for your ancestor.
For Ellen McCue M ditto, I found 0 M rcds for this name, or any of your variants. You MIGHT confirm the name spelling by viewing all of the M rcds for William above.
For Mary Monahan B 1844 (exact) somewhere in Ireland with the father's forenamed William, I found 0 rcds.
For Thomas Monahan B 1837 +/- 1 yr ditto with a father's forename Thomas, I found 2 rcds: Leitrim, Longford & Galway.
For Ann Monahan B 1840 ditto, I found 1 rcd in Cavan.
For Peter Monahan B 1842 ditto, I found 1 rcd in Co. Longford.
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.