Caveny - Ballina

Hi, it seems my family came from Ballina, Co. Mayo. Their family name was Caveny (though we have numerous spellings of this name!) My 2nd great grandfather was Patrick Caveny, born in around 1844. Through census returns in England I have determined his parents were Thomas Caveny born around 1806 and Ellen born around 1811. He had siblings Michael born around 1838, Thomas born around 1845 and James born around 1854. There may have been more that didn't move with them to England, where my branch ended up in Tadcaster, Yorkshire. The only clue I have to where in Co. Mayo they originated is when on the 1881 census Thomas Caveny (jr) put his place of birth as Ballina. I have tried to search the records on NLI website without luck. Unfortunately there is a gap in the baptism records in 1844 when Patrick was born. I'm going to try these records again, page by page. After 19 years searching, I'm getting desperate to break through this brick wall! Thanks for taking time to read this, kind regards, Cheryl.