Gleeson in Ballyhaunis

Thanks to help via the website I now have a couple of additions to my family tree. It's narrowed to Ballyhaunis. Andrew Gleeson (sometimes called Henry) born in 1794 married Mary ? born in 1805. I have some info that she may have been Mary Byrne? So if anyone knows of a connection there. They had at least one child, James Gleeson born 1832 and married to Bridget Haddican in England.

I'd like to find out Marys surname if possible and if there's a connection to the Haddican family in Ireland.