I'm not disputing that the Thomas White of Nottingham is attached to incorrect parents, though I haven't actually investigated anything.
I am though disputing that the book by Almira Larkin White is well-researched, at least the information on pp. 11-12. It doesn't actually cite any sources and although there is a will of an Agnes White dated 1549, who names Thomas White, there is no relationship mentioned with him.
The only family mentioned that I can see in a quick skim is "To John Hyllarde, my lytell boye, 2 silver spones". As there are a number of Hilliarde mentioned as receiving the residue of the property and also as executors, it seems more likely Agnes White is related to them.
I can't see the 1551 will of Thomas White, though that doesn't necessarily mean it doesn't exist, but if a replacement profile for Thomas White is created, then I would recommend that sources are found that confirm the details found in White's book beforehand.