Lower, Mark A (1860) Patronymica Britannica: a dictionary of the family names of the United Kingdom. London: J.R. Smith. Public Domain.
WOODGATE
As described in historical dictionaries
Patronymica Britannica (1860)
WOODGATE. From residence near the gate of a wood. An ancient Kentish name, occurring in the form of Ate-Wodegate, temp. Hen. III. In documents of the XIII. centurj, the name is sometimes oddly written Wdegat or AVdegate.