I found Frederick Southgate with his parents John and Elizabeth in 1851 and 1861 in Kirby-le-Soken. John born around 1824 Kirby so he may well be the John Southgate 15 (rounded down) with James Southgate 35 (likewise) in Little Bentley in 1841 which is not so far away. Birthplaces only "in county" of course.
However although James Southgate the presumed father of William James is consistently born Colchester, the only entries on which he can be found with his presumed parents have them as Thomas and Martha, he born "Hadleigh, Essex". This is down near Southend and a long way from Colchester, and I wonder if Hadleigh, Suffolk is meant - this is a little nearer and near Martha's birthplace of Nayland, Suffolk which is even nearer to the city. The family were in Bow, London in 1861 but in Colchester in 1851.
He is probably the one born Colchester 1846 with the MMN Carter, but there's no such marriage between a Thomas and a Martha. There is a possible one in Thingoe (Suffolk) in 1840 (William S and Mary Ann C and another 6 folk on the page) and another rather late in 1846 where there's a Thomas and a Mary Ann (same caveat). Tantalisingly, there's a Thomas marrying in Colchester in 1840 where there is only one bride on the page (and two grooms) But a marriage search on that quarter reveals no Carter or Martha in Colchester. Anybody else fancy a go?