I'm at the library, looking at Georgia's Black Book. I find several Haywoods, but no John or James. Most are black, one is mulatto.
The names found in volume 1 are Ike, Black; entered prison 30 Jan 1876, convicted in Lee County. I'll abbreviate the rest.
Tom, Bl, 29 Apr 1876, Sumter
Issac, Bl, 30 Jan 1876, Lee
Joseph, Bl, 15 Dec 1873, Cobb
Yook, Bl, 17 May 1877 Camden
Bosey, Y, 9 Feb 1875, Chatham
Ida, M, 8 May 1884, Chatham
Volume 2 has two Johns, a J.H. and a Josiah.
Bose is listed under prison deaths. No dates, but from Chatham.
Under requests for pardons/parole is:
J.H., Lamar County
In Chapter IX, Reports of Racial Violence, page 154
Petition of Lydia Braughton (black) for clemency for her son King Braughton, convicted of murdering Henry Screven (black) in Chatham County. It is signed by many, including John H. Haywood and Josiah Haywood.
There are more Haywoods in the book, but I think the John H. or Josiah could be what you are looking for.
At least it narrows your search to Chatham County, GA.
Good Luck! Do let us know what you find.