Welcome to the society - I hope that you will be able to get some help to further your research.
I see from a transcription of Bristol Parishes available from the Bristol & Avon FHS that Benjamin CARTER was a widower when he married Ann RIGHTON - Have you found his first marriage?
There is a christening of a Benjamin CARTER s/o Benjamin & Mary CARTER 31st Aug 1823 at St James Bristol which just might be a son by his first marriage. There is a burial of a 26 year old Mary CARTER at St Mary Redcliffe, Bristol on the 5th Jan 1827 which might be her death. No actual proof of the connection but they might be worth looking at.
You say that Benjamin CARTER was a Journeyman Confectioner - you must have found that somewhere other than the 1851 census because on that census he is simply a Confectioner. However on a CD of Bristol Apprenticeships I did find that he was apprenticed to a Samuel LUCAS, Confectioner, on 26th June 1809 giving his father's name as James CARTER, a cordwainer (Shoemaker) from Chew Stoke. As a result of completing that apprenticeship Benjamin CARTER was admitted as Burgess (Freeman) of the City of Bristol on the 11th June 1818.
I can't immediately see Benjamin's baptism at Chew Stoke in our transcriptions - you may have to look in surrounding parishes but at least you have a name for his father now.
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