Lorraine
I think we may well be connected but your William is a bit later than my research. I assume this is him from the IGI

1. WILLIAM HARTSHORN - International Genealogical Index
Gender: Male Christening: 08 JUN 1794 Polesworth, Warwick, England

Father William, mother Mary

I do not have a marriage for William and Mary but as a guess the marriage would be about 1792 implying William the elder was born 1770 ish. There is a William Born 25/11/1770 in Polesworth to William and Dorothy who started having children in 1763 implying a birth of this William about 1740. The William on my tree was born in 1747 but 16 seems a bit young to be having a family so I think they must be different Williams.
Alternatively there is a William born in 1731 (son of George)who is a more likely age.
Have you tried the 1841 and 1851 census with different spelling eg Hartshorne/Hartson?

Richard