Here I am again.
The Gainsforth obituary I finally found, helped by people on here, gave me this:
Thomas Gainsforth born June 1st 1832 Ireland.
Arrived New York 1842 and settled there and in New Jersey until 1856 when he went to Wisconsin.
I have him in Wisconsin and from there till his death in Nebraska in 1920. It's that new info which I thought was going to be easy, but it's not.
I've been through ships' lists, NY and NJ 1850 censuses, and what I can get of NY naturalisation records but to no avail.
If 1842 is right he may have been with his parents (Richard and Sarah??) or with another relative.
I'd be grateful for any help. The name is difficult (different on every census I have!) but on the other hand it's not common!