Trish,
This looks to be Patrick & Annie’s marriage in Crossmaglen, Co Armagh in January 1921:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1921/09215/5333546.pdf
I think this is Patrick in the 1901 & 1911 censuses too:
http://www.census.nationalarchives.ie/pages/1901/Armagh/Creggan_Lower/C…
http://www.census.nationalarchives.ie/pages/1911/Armagh/Creggan_Lower/Cornoonagh/333062/
Cornoonagh is in Co Armagh and from 1.1.1922 onwards that became part of Northern Ireland. Assuming that’s where Patrick and Annie lived after they married, then the birth records you need are recorded in GRONI. It’s the only repository for birth, death and marriage records in what is now Northern Ireland, 1.1.1922 onwards. (Ancestry and other websites don’t have them). You mention that Cornoonagh is where your mother was born so it does look as though that’s where they stayed.
Births in Northern Ireland within the past 100 years, marriages within the past 75 and deaths within the past 50 aren’t searchable on-line. They are open to the public, just not on-line. If you know all the key details eg exact date and place of birth, parents names etc, you can order a copy of a certificate from GRONI for £15. https://geni.nidirect.gov.uk
But if you don’t have all the information, you need to get in touch with them to get a search done. They will do a 5 year search for £7 and if they find a matching certificate you can purchase it for £8, coming to a total of £15. It will be posted to you a few days later. You pay over the phone using a credit card. You type your number in on the phone keypad. They don’t know what it is. They won’t deal with these searches and orders by e-mail because they need you to type the credit card number in yourself.
Phone: 0300 200 7890 (+44 300 200 7890 from outside the UK)
You know that one of the missing births was James in 1931 so he shouldn’t be to difficult to trace. For Michael, you will need to ask GRONI to search from say 1923 to 1933 and if necessary, a few years beyond.
There’s a report in the Drogheda Argus & Leinster Journal of 13th July 1963 about a Peter Woods of Cornonagh who had been arrested for drink driving. He admitted to having had 5 bottles of stout. The Magistrate gave him the benefit of the doubt and he got off the charge. So the Woods family appear to have still been in that area in the 1960s. Presumably Peter’s departure for the US was after that period.
