Do you know where in Cork Cornelius Mullane lived? Have you searched the civil records for his death on irishgenealogy.ie? Did you look for his birth or the marriage in the church records https://registers.nli.ie/
, presuming he was RC? I had a quick look at the Mourneabbey register but didn't see him between 1860 and 1875 so you might have to search neighbouring parishes.