Hi Laura
Really pleased that you replied.
I think that I've solved some of the mystery.
As I said, the key was finding James Farrar (John's father) in the 1851 census.
The information there is
James Farrar 43 - Copper Smelter
Ann Farrar 47
James Farrar 19
John Farrar 18
William Farrar 16
Mary Farrar 13
Edward Farrar 10
Henry Farrar 7
This in Eccleston, St Helens, Lancashire.
Then using this, going back to the 1841 census, we find
James Farrar 30
Sarah Farrar 30
James Farrar 10
John Farrar 9
William Farrar 4
Mary Farrar 2
Again in Eccleston
So John's mother was probably Sarah, who then died between 1841 and 1851, James re-marrying Ann.
If you then start to look for John Farrar in St Helens, you find a Christening record for 10 Feb 1833 for John Joseph Farrer, parents John Farrer and Sarah.
And here we are!! So the information in the 1871 census for John Joseph is correct, but he somehow changes his birthplace for the 1881 and 1891 censuses.
I can't find John in the 1861 census, but I haven't given up.
Hope this helps!
Kind Regards
Howard