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 Winifred Noon B 1837 +/- 4 yrs somewhere in Ireland with the father's forename unknown, I found 4 B rcds: 1834 Roscommon, 1836 Roscommon, 1837 Galway, and 1838 Roscommon. You will need a parent name, more precise DOB, etc., to select which, if any, of these is for your ancestor.
For Martin White B 1833 ditto, I found 11 B rcds: too many to list here. Will leave this to you.
Suggest you search U.S. sources such as census and civil/church rcds to find their DOB and/or parent names.
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.