What seems to be well established is that John Bassett (1518 - 1541) and Frances Plantagenet (1519 - 1568) had a son named Arthur Bassett born in 1540 and died in 1586, but no evidence that suggests there were any other sons.
So far there is no evidence to prove that this John Bassett was also a son of John Bassett and Frances Plantagenet, nor any evidence suggesting other candidates to be his parents.
It seems likely there is a common Bassett ancestor, however, this ancestor may have preceded John Bassett (1518 - 1541) by several generations.
Genetic genealogy would be of great value here, particularly if the DNA of descendants of Arthur Bassett, the known son of John Bassett and Frances Plantagenet could be compared against descendants of this John Bassett.
If anybody has come across any documentary evidence of other sons John Bassett and Frances Plantagenet had it would be most useful if you might upload that here for the collective benefit of all.