I received the marriage certificate and Richard is Joseph's father! So him living back with Richard and Mary in 1871 is the correct person... either that or their's another Joseph with a father called Richard!
As a double check, were the occupations as recorded on the marriage cert the same or similar to those recorded on the censuses for Joseph and Richard?
As for finding out what Joseph's mental health/injury issues were, you may find something in the Gale Newspaper Archive (probably free through your local library service) or the British Newspaper Archive (pay to view, but the free search facility can be quite illuminating). If Joseph behaved oddly, he may have got a mention in the newspaper. The chance of finding proper medical notes is about zero, I'd say.
When you get your time machine, please can you let me know? 
Joseph's dad does, Richard... It says he's a labourer and that matches the census entries... Hannah's dad Ralph though... I'm not entirely sure what it says! The marriage cert scan that they've given me is very dark and the only letters I can make out are "Pat" I'll have to have a look at an occupation list for possible matches, failing that see if someone can decipher it! I've not looked at Hannah's side of the family much though so I'll see if I can find the census entries for him!
Thank you for the tips on his possible mental illness. His mark as an Imbecile could've even come as a result of Hannah leaving him! (oh the speculation about this marriage is rife in my head
) But it could also have been a contributing factor to Hannah leaving him, either way, he definitely isn't dead before she re-marries, so something must've happened for him to go back home and for her to take the children with him.
Don't worry about the time machine, as soon as I've found one, you guys on here will be the first to know and use it
