I really hope this reaches you as I noticed your posting was ages ago.
I have a John Finan,widowed, aged 80, living in Liverpool on 1861 UK census. With him is his daughter Catherine Finan aged 20 and his son John Finan aged 18.
John (junior) goes on to raise a family of his own (Charles, John Thomas, George, William & Samuel) then in 1891 he left his eldest son & family and went with his younger unmarried sons to live with another group of Finans. As they were all coal miners, I’m assuming they moved to wherever there was work available.
These Finans are made up of more than one family, all living in a big tenement block near Barnsley, but amongst them are 3 teenage boys Henry, John and Richard who I suspect may all be sons of John's sister Catherine.
By 1901 John Finan is not at that address (may have died) but Catherine is there herself, now aged 57.
I wonder if there’s a connection here because of the continuation of the names John and Richard? Could my 80-yr old John be either your John Fynan, or his son?