There is a baptism for a Henry Randall Pelham on 16 Mar 1806 (dob 12 Feb 1806), the son of a Joseph Pelham, and wife Elizabeth. Baptism was in St Martin-in-the-Fields, Westminster.
Six years earlier, on 9 Feb 1800, also in St Martin-in-the-Fields, Westminster, a Joseph Pelham married an Elizabeth Keede.
It looks like they also had an earlier son than Henry, that being Joseph Walter Pelham, bapt. 04 Jan 1801, also in St Martin in the Fields, Westminster, Middlesex, England.
Also, Henry Randall was first married to Hannah Marshall, in 1830 (although the banns were in 1827... seems odd!)