The IGI has a William Welham marriage to Mary Piper at Offton Suffolk on 31 Jan 1819, with baptisms of a number of children with parents William and Mary in various parishes in Suffolk, including the Richard that Annette found in Newmarket. Whether these are all the same couple I can't tell. Some were baptised in Offton, others in Nettlestead. They include Susannah baptised in 1836.
Living in Offton in 1851 was Susan Welham 16 b Nettlestead, shown as daughter of William and Mary MILLER. The mixture of surnames imply a second marriage for possibly both. There's a marriage in the Mar quarter 1851 in Bosmere registration district which includes both Offton and Nettlestead, where two of the names are William Miller and Mary Welham. There are also deaths of William Welham in Bosmere reg district in 1845 and 1848 either of whom could be the one.
In 1841 in Nettlestead were William Wellham (transcribed on Ancestry as Willham) 45 ag lab, Mary 40, William 20, Hannah 14; John 12; Betsey 10; Susan 6; Sarah 3. All b in Suffolk. But no Richard who at the age of 10 should have been with his parents. So is this Offton family a different one to Richard's? I think it may well be. Offton and Nettlestead are only a mile apart, whereas Newmarket is 38 miles away.
So having probably ruled out this family it still hasn't found Richard and his parents William and Mary, if still living, in 1841. Why did Richard move to Wales from Newmarket at some time before 1851, and was it with his parents or on his own?
So to date we haven't found a marriage of William Welham and Mary, presumably the mother of Richard, nor a second marriage of William Welham to an Elizabeth Carter although I'm still dubious about this on the basis that Welsh marriage certs don't give mothers' names.