Hi,
Thank you for everyone's reply, I'll answer each:
JohnHood - At present I am thinking either William or Jane are candidates to be Henry's father. Henry named his first child William and his second child Jane so this is not very helpful. I was thinking that if Jane was the mother then there may be a bastardy bond.
Klouman - Yes, William married Pheobe in 1796, this was his second marriage. Jane above is from his first marriage. William was having children with his second wife Pheobe at the time that Jane had one child that I know of. I'm thinking that either William and Pheobe or Jane and unknown father may be Henry's parents. However I can't prove either yet and could be on the wrong track.
Willsy - Now the bastardy bond for Jane is interesting given it mentions a Henry Watson as the father, makes one wonder if my Henry Leeson might have been named after his father. I know that Jane had one child out of wedlock at the beggining of 1815 so the bond may be for this other child. However this is definitely a record that I will check when the Leicestershire archives open.
Thank you everyone for your assistance, your replies have been great.