The Barbour Collection of Connecticut Town Vital Records (NEHGS, Boston, 2011) Vol. Branford, Page 185: Birth record of his children.
"Connecticut Births and Christenings, 1649-1906", , FamilySearch (https://familysearch.org/ark:/61903/1:1:F74Z-5MK : 7 January 2020). Shows the birth of May Way, a male child, to parents Thomas and Ann Way on 15 May 1702 at New London, New London, Connecticut, British Colonial America.
Families of Ancient New Haven, Vol. VIII [1] via Ancestry.com (probably available on Archive.org) shows the birth of May Way 15 May 1702, died about 1769 at (maybe Bethlehem, CT, married Abigail_________ (no last name); shows many children including Child Joseph (born 19 Nov 1742), who married Philomela Bradley.
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DNA Connections
It may be possible to confirm family relationships with May by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
However, there are no known yDNA test-takers in his direct paternal line.
Mitochondrial DNA test-takers in the direct maternal line: