There's a Mr John Woodham of Gastlings buried at Southill on 14 Jan 1771, and a William Woodham, yeoman, on 4 Aug 1784 (possibly the one who married in 1771). I can't see an obvious connection between the Southill Woodham family to the Gamlingay one.
David. Even though I don't have access to 'the file' I think I can answer the above query. I found the following on A2A (Access to Archives) whioch is now part of The National Archives site. The original document is at BLARS.
BLARS Ref: HF13/5/12 Bargain and Sale dated 7 Feb 1767 mentions
Parties: 1. John WOODHAM the elder of Gamlingay, Cambs, gent
2. John WOODHAM the younger of Southill (son and heir of above)
Property: 4 acres of arable land in the fields and bounds of Clifton and Henlow some time purchased by 1. from Mary KING.
A part of the same document quotes John WOODHAM of Little Wild Street, Lincolns Inn Fields, Middlesex, coachmaker (nephew of John WOODHAM late of Gastlings, Southill, gent, deceased) as vendor of the above property 14 & 15 Oct 1771.
According to A2A BLARS has at least two other documents mentioning John WOODHAM of Gamlingay. The first is dated 7 Feb 1756 when John WOODHAM acquired a mortgage of £70 from Thomas THOMPSON of Potton, barber and peruke maker, for a cottage or tenement with orchard and appurtenants at Wilden in the occupation of George FITZHUGH. BLARS Ref X631/17. Evidently John didn't receive the £70 so the property was vested in him and the second document (BLARS Ref: X631/21) refers to its subsequent conveyance.
There may also be other documents relating to these families on A2A but the following , held at Somerset Archives could be interesting.
1797 George, Earl of Macclesfield leased two farms in Gamlingay, Cambs, to Willam WOODHAM of (part of document missing) in Cambridgeshire for 12 years.