There is simply TOO much Bad Information out there MIS-connecting all sorts of people to these hoity-toity recusant Catholics.
A lot of it comes from Emma Siggins White and her attempts to gentrify her husband's ancestry (he came from a line of Somerset shepherds). "Genesis of the White Family" is at *least* 50% pure codswallop, not least the assumption that anyone and everyone who ever used the surname "White" MUST belong to the same family (oh no they don't!).
Also, he has extra sons (even without a recent rejected merge) and is missing several daughters.
According to the Visitations of Hampshire, Robert had only two sins: the extremely prolific Sir Thomas White, Sr., MP, for all the good it did him, and a Henry who died heirless.
No Robert III.
No Edmund.
No William.
There were, however, three more (unnamed) daughters.. One married a Webb of Salisbury, one married a Yonge of Berkshire, and one married a Normanvile of London.
The Visitations aren't 100% accurate (and some are worse than others), but the errors in this case appear to be of omission (first names of the three daughters and their husbands) rather than commission.