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John Burke is a very common name in Ireland especially in Mayo and Galway. Also, many RC parishes in those two counties do not have records back to the time period around 1822. Having parents names would be essential.
I did search on the subscription site Roots Ireland and they had 53 John Burke baptismal records from 1817-1827 in Mayo and Galway. I searched for a John Burke/Winifred McDermott marriage record with no success and did not find a baptismal record for a Patrick Burke with father John and mother with the surname of McDermott.
Likely the family was from a parish which has late starting records beyond 1850.
You may want to add John Burke to our XO Chronicles site and possibly someone will see a connection. https://irelandxo.com/ireland-xo/history-and-genealogy/ancestor-database
This IRO Insight may be of interest https://irelandxo.com/ireland-xo/news/irelandxo-insight-how-dna-can-help-you-connect-ireland