Hi Lorraine
Have you had a look into this family at all?
1861 census 37 Prospect Street, West Derby
RG 9; Piece: 2727; Folio: 85; Page: 19
William Potter 52 b ireland
Eliza Potter 50 b ireland
Copfield Potter 24 b ireland
Eliza Potter 20 b Liverpool
William Potter 17 b Liverpool
Sarah Potter 14 b Liverpool
Thomas Potter 10 B Liverpool
Benjamin Potter 8 b Liverpool
William Driver 8 b Birkenhead, nephew
Sarah Driver 3 b Birkenhead grandaughter
Jeremiah Driver 6 b Birkenhead nephew
I think the Potter's were in Manchester in 1851, but birthplace's are Manchester for the children in 1851 and William Potter born Lancashire not Ireland. but I think it's the same family because Cawfield is there and a couple of the other children. Maybe this is where the confusion with birthplace comes in, if he was with a family from Manchester, he was born in Cheshire, but lived in Liverpool!

nobody seems to know where they were born

Also wondered if Henry b c 1877, may have taken a the name Jeremiah from an Uncle, one of my ancestors did that
1851 census
HO107; Piece: 2225; Folio: 704; Page: 4
Oldham Street, Mancahester
William Potter 42 b Lancashire
Eliza Potter 42 b Ireland
Cawfield Potter 18 b Manchester
James Potter 16 b Manchester
Mary Potter 11 b Manchester
Eliza Potter 9 b Manchester
William Potter 6 b Manchester
Mary I Potter 4 b Manchester
Thomas Potter 1 b Manchester