I missed the notification of your reply! Darn it.
Findagrave lists Henry's birthplace as New Jersey, and a Henry Drake did serve in Capt. Joseph Stull's New Jersey Company, raised to quell the Whiskey Rebellion in Pennsylvania in 1794. They marched into Western Pennsylvania, altho likely not as far north as Erie County. Still, if this was our Henry, he may have then formed the idea of resettling in the area.
Of course no census while Henry was alive contained birthplace information. It's not until the 1880 census that his children cite a New York birthplace for him, more than 100 years after his birth.
Two of Polly's sisters married men from Connecticut, and the Godard parents, Aaron and Lorohamah, were from Simsbury, Connecticut. We don't know Lorohamah / Ruhamah's last name or anything else about her ancestry, but after she was widowed she was "warned off" —encouraged to return to Simsbury rather than become a financial burden on the town of Middlefield, Mass.
It was at about this point that her children moved north and west up the Hudson and Mohawk river valleys as so many were in this period. Only Aaron and Alice stayed in New York for any period of time; Anson, Polly and Desire and their spouses headed for Erie County. We have found no further trace of Lorohamah or of the fourth daughter "Rumcy" (likely named for her mother) who married an unknown Mason.
I don't know anything about the Cheesemans —but there appear to have been quite a few Benjamin Cheesemans... in New Jersey. I see on the map from 'A history of the town of Middlefield, Massachusetts' that the Cheeseman property was downriver from the Godard's on Factory (Blush) Creek.