If Jonathan Fay is connected with the Fays of Marlborough, MA he was not Huguenot.
There is a line of FAY that comes from Ireland and entered New England from Canada.
If Jonathan is tied to the Green Mountain Boys look into the Stephen FAY line of Bennington, VT. This line dates to 1762 and the establishment of the Republic of VT. Green Mountain Boys lived in VT. Lyman could not have been part of the Green Mountain boys by virtue of his birth year. Green Mountain Boys were active during the Revolution; Lyman was born after the war ended.
I find no Patriot Jonathan FAY through the DAR. This doesn't mean he wasn't in the Revolution, just that no one filed an application on his line and is accepted. He is the only likely candidate to be a Patriot of the two. Land in NY was offered several men who fought in the war. Try the National Archives.
Look for land deeds in Bennington, VT as well as in NY (if available).
Several FAY lines went to NY.
I located a Jonathan FAY in Brimfield, MA. He married an Anna. Their children born in Brimfield predates the date of your Lyman. They are Lucy 1769 m Rufus Bates; Esther 1771 m Joel Barrows; Anna, 1774-1776; Joel 1778; Jonathan 1780; Anna 1783; Simon 1786.
The FAY line I traced through Brimfield, MA also went to NY - Cazenovia, Madison County, NY. I know of no connection to Lyman to FAY genealogies I am aware to date.