Hi Rap
As others have said, a marriage certificate for Henry, proving whether or not Henry's father is Benjamin, is essential, as it's no use building a whole dynasty on suspicions!
Having said that, I think your suspicions look good. You are fortunate that the name KERGAN is rare in the West Riding- I suspect it is parish clerks' attempts to make sense of an unfamiliar name being said by people who were illiterate and therefore not able to offer any guidance on spelling. I would keep the Irish names KEEGAN and GEOGHEGAN (pronounced more or less Gaygan) in mind when searching.
In case it should be useful in future, when you have definitely established Henry's parentage:
The IGI marriage at Halifax St John's between Benjamin KERGON and Hannah BAIRSTOW is correct. Both are of the parish, and Benjamin is described as a cloth dresser.
The IGI christening 13th June 1824 for Benjamin KERGON, son of Benjamin and Hannah is correct. Father Benjamin is described as a fustian cutter of Halifax, but that fits OK as a fustian cutter is a kind of cloth dresser. Benjamin junior was born on March 24th.
Benjamin is the couple's second child, the first being Margaret who was born Nov 8th 1821, christened 3rd February 1822. I don't have the records for the possible IGI christening of Benjamin senior at Square Chapel 31st March 1793, son of Patrick and Margaret, but can confirm the marriage of Patrick KERGAN, gardener of Halifax and Margaret Sutcliffe, widow on 19th May 1783. Neither bride nor groom was able to sign the register.
Regards,
Lesley