I see that Carrickmacross is shown as his birthplace on a military record.
If he was Catholic, the records begin in 1838, so you will not find him there.
https://registers.nli.ie/parishes/0292
Having written the above, I can’t see a military record for Patrick with a birthplace. There is one for an older man so I wonder if I made a mistake.
Lots of trees show Carrickmacross but again, I don’t see any evidence. Similarly, they show him in Manchester in 1851.
There are 3 Patrick McCabes in Liverpool in 1851 where he enlisted in 1852.
What evidence do you have for his birthplace?