I had a go at searching for Michaels birth on RootsIreland and it appears to be there, Michael 1821 Drogheda, father John, but when I put in Catherine McCarthy I lost it, however it came back when I left out the name Catherine but kept McCarthy.
Once you have the parents you can take out the child's first name and put 0 in its place and it will give you all the children for that couple, but there was only that one result, so from that he appears to be an only child. Which you probably already knew.
I than tried to find a marriage for John and Catherine but no luck.
No help at all I'm afraid.