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 Cornelius O'Callaghan B 1823 +/- 4 yrs in Co. Cork with the father's forename unknown, I found 19 B rcds. NONE in Glanworth parish, but 3 were in Mallow: 1820, 1825 and 1826. One of these, if any, MAY be for your ancestor. A view of the detailed rcds MAY reveal info to verify it.
You have some of this ancestor's military service. Suggest you contact the UK National Archives for his complete dosier. It would be an interesting tidbit to pass along to your heirs.
Co. Waterford rcds aren't in the database yet so I can't search for the M rcds. Suggest you wait until they are. Alternatively, you can contact an LDS Family History Center and view its microfilms of the original parish registers.
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.