Hi Libby,
The 1841 census has a John Bingham, 47, ag lab, living with wife Bridget (43) and 8 children in Pattiswick, Essex. Is he the one you have in mind? Children are George, Eliza, Mary, Hannah, John, Stephen, Ephraim and Sophia.
Edited to add: if he is the right one, he's dead by 1851 as Bridget is widowed and head of household in Pattiswick with 3 of the children (Ephraim, Sophia and Stephen).
Anna