I will try and find a connection for John HH to Elizabeth Hagarty (the name Elizabeth does feature in previous and future generations), there may well be a link.
If you go to Ancestry.com I do have a tree on there (MANY LEAVES). It contains most of my Hawkins/Hagarty folk although they are partly unconfirmed. The tree is accessible for free public viewing. Follow backwards from the Home Person -R C Unwin, Sutton, Thornbury to Hawkins.
Just out of interest, a charactaristic of this branch seems to be that they have a sail shaped (profile) nose! My father has one, his great grandmother being a Hawkins.