I have done quite a bit of research in an adjacent county in Georgia. There are several families in this Georgia county with unwed mothers during this same time frame.
Sometimes, the death certificates for the children who were born out of wedlock will state the mother's given name and list the biological father's last name as surname. I guess it was an attempt to hide their illegitimacy?
So you might want to consider that your John White is the son of Rebecca White and an unknown Mr. Griffin.