There are no Henry Budge events for Bedfordshire on the IGI, which for this period is just about complete for baptisms and marriages. But you'd have been better posting it on the Huntingdonshire board.
But I see that you've posted this request twice already on the Hunts board at
http://www.rootschat.com/forum/index.php?topic=702601.msg5455534#msg5455534 and
http://www.rootschat.com/forum/index.php?topic=745038.msg5921441#msg5921441 and I really can't add to what I've already said. He didn't marry an Elizabeth in Hunts between 1701 and 1740, so presumably they married in another county.
But I'm concerned about the birthplace of their children. You say Ann was baptised in Godmanchester (what's the source for this?) yet the IGI, extracted so reliable, shows it at Glatton. And a number of other children of Henry and Elizabeth were also baptised at Glatton - Elizabeth in 1712, Henry in 1714, Thomas in 1722, Hannah in 1727, Jane in 1734, and then there's Susannah in 1725 whose parents are shown as Henry and Mary yet you listed her as a child of Henry and Elizabeth. Are there actually any baptisms at all in Godmanchester? Have you checked Glatton and Godmanchester parish registers?
Are you sure the Glatton and Godmanchester families aren't two distinct couples? Glatton is 15 miles from Godmanchester.
If the family were from Glatton rather than Godmanchester then it's a whole new ball game, as they could easily have married in Northamptonshire if Elizabeth was from a village on the other side of the border, as the county boundary is only a mile or so from Glatton. I' d try a lookup request for the marriage on the Northants board
Small inaccuracies can have significant repercussions!
David
Which web sites record his baptism in Hunts around 1690? If you're referring to online trees most are completely unreliable. And the Budge trees on Ancestry are particularly so and should be ignored. Many show Henry born c1690 as marrying Susannah Griffin in 1736, when Elizabeth his wife was still living, whereas it looks certain that the marriage was of Henry's son Henry baptised 1714.