Doing some more cleanup on this and the Timsbury lines. Ancestry of Mark (Marcus) White of Timsbury is undocumented, as is the name and ancestry of his wife.
While Mark White *may* have had a brother John who was associated in some way with the school at Winchester, he was not and could not have been the same person as John White, **Bishop** of Winchester. That one belonged to the Farnham/Swanbourne/Hutton nexus and was the possibly-twin brother of Sir John White, Alderman and Lord Mayor of London. (Both brothers John redesigned their personal arms with closely similar blazons, underlining the relationship.)
A conundrum remains as to the ancestry of Sir Thomas White, first White of Tuxford. Nottinghamshire History (the site has a bad Certificate, use at your own risk) claims him as the son [or grandson] of Johannes White de Colyngham; against which must be set that the Visitations of Nottinghamshire and Burke both state that he was the son of Nicholas White of Suffolk. (The Colyngham/Southwick/Tuxford Whites are said to have originated in Suffolk as distant cousins of the Farnham/Swanbourne/Hutton Whites via remote ancestor Bartholomew le Wite of Suffolk.)