Statutory birth, death and marriage registration (in some jurisdictions called Vital Records) only started in Ireland in 1864. Prior to that you need to rely on church records (where they exist). I searched the on-line RC records for a John McKenna = Bridget Gormley marriage but did not find one. Likewise I searched for a baptism for Patrick 1856 – 1861 to same parents but without success.
I also searched the statutory birth records 1864 – 1873 to see if a couple with those surnames had any children in Northern Ireland in those years. There were none.
The RC records for Omagh (RC parish of Drumgath) are on-line and have been indexed. I searched them for a baptism for Patrick. Nearest I found was on 19th Dec 1858 to John McKenna and Mary Toner.
The John McKenna married to Mary Toner was a Cooper by trade and lived in Castle Lane, Omagh. They had several more children and were still in Omagh in 1868.
If the parents names were John McKenna & Bridget Gormley then I suspect that their marriage and Patrick’s birth were in a parish that doesn’t have records for the relevant years. That will make it difficult to trace him. This link takes you to all the RC parishes in Co. Armagh. If you click on each one you will see what records exist: