A possible scenario.
William Henry is the illegitimate son of Sarah Ann Philpott. She in turn is the daughter of George and Sarah Philpott baptised in Chilham on 4 September 1836.
Baptism at
https://familysearch.org/pal:/MM9.1.1/N188-QZNThe 1871 census for Faversham RG10, Piece 979, Folio 101, Page 26 has
William Hampshire 30 born Ospringe
Sarah Ann Hampshire 34 born Chilham
>>William H Hampshire 11 born Chilham
Edith Hampshire 6
Jane Elizabeth Hampshire 6 months
Sarah Ann Philpott married William Hampshire in Faversham District in the March Quarter of 1863 Vol 2a page 822.
However, the only satisfactory way forward it to get hold of (buy if necessary) a copy of William Henry’s Marriage Certificate to find his father’s name, if there is one on the certificate.
Hope this helps.
Regards, Tony