I suppose the logical place to start would be to check the records of Huntingdon Assizes in the year or so prior to March 1787 to find more information on his conviction. These would be held by Huntingdon Records Office.
The 1776 Payne looks a bit young to have been transported in 1787, but there is another Pain on the BVRI christened at Kings Ripton on 26 Feb 1748, son of William and Elizabeth. William Jacob married Elizabeth Pain in Kings Ripton in 1737 (Hunts Marriage Index), so this, it appears, is where the christian name "Pain" derives. This may have been the Payne of Houghton cum Wyton who married Elizabeth Hollis in Godmanchester in 1778 (also on the IGI as an extracted entry)
I wonder if Henry might also have been a son of William and Elizabeth to have named one of his sons Payne as well?
Regards
David