I didn't see a mention that Matthew and Bridget were not first cousins in the thread but early on was this-
Yes that's the Mathew Cullen. I have got back to his parents by looking at the Irish Genealogy site (complete with Latin names!) and found several of his relatives. I did find a record for a baptism of a Brdget Cullen, but the parents seem to be Mathew's uncle and his wife, Elisabeth Roe (who lived in Dublin City for a while but also in Barnacullia) ... I don't think first cousins can marry under Irish and Catholic Church law, so am assuming I had found the wrong one?
Which is why I posted that first cousins can marry.
I know that you (Tara) had found Bridget in the 1901 census (reply #9) but her age being 17 not 11 wasn't mentioned in that post.
Nannyj- no need to quote all of previous reply as it makes you post very long

Before our current health system the Workhouse would have been the place for hospital care so that's likely why wee Matthew ended up there sometime after his mother died.
https://www.workhouses.org.uk/Rathdown/ (you can see that the hospital building is show near the bottom of the map)