My sister-in-law has uncovered the following:
Mary Ann Clapham who married Henry Cautley in 1825 was probably not related to the Claphams of interest. She was born in 1801 in Burton Pudsea to parents who, curiously enough, were named William and Sarah Clapham. However this William was a yeoman, born in Burton Pudsea in 1765. He married Sarah Harland, same age as her husband, also of Burton Pudsea, in 1787. They went on to have a number of children between 1789 and 1801, Mary Ann being the last.
Henry Cautley was born in Dunnington Yorks in 1798. At the time of his marriage in 1825 he was living in Hedon, Yorks, and practicing as a surgeon. He died in 1874 in Bramley, Yorks. He and Mary Ann Clapham had at least two children, one of whom was also named Henry (Henry Strother Cautley).
Unlikely the Henries and James Lambert Clapham moved in the same social circles. The younger Henry wasn’t born until around the time JLC went to NZ (if that was 1862-ish).
The only possible connection is if the older Henry (who married Mary Ann) knew the Thorney Claphams through some shared medical training/apprenticeship. Even then I don’t know why JLC might have named a child after him. the age difference is too great, for a start.