We didn't have the capacity to annexe Ireland. However, you also said England did not exist until at least 500 years after Patrick lived - we know that's wrong because Bede wrote a History of England and it's people in the early part of the 8th century.
No he didn't. He wrote the Ecclesiastical History of the English People, no mention of an England as a nation or state because there was none until Athelstan. There was no England, just Bernicia/Deira (and then Northumbria), Mercia, Wessex etc.
Bede's 'English People' was concerned with the Anglo-Saxons, and large chunks of what we call England now weren't even held by the Angles or the Saxons in Patrick's time given it was a hundred years before this map where even a century later less than half of 'England' was held by the 'English'.
No one thinks there was an England in the 400s.
Logon or register to see this image
B