The John Cooke or Cox described here with wives and children has different origins. This is an apparent conflation and as such, he should be removed from the parents who are presently shown.
John Cooke who was the son of Philip Cooke and Elizabeth (Belknap) Cooke had only one son, Sir Anthony Cooke (1505/6-1576). His vital dates were (1485-1516), and his alternate spelling was Coke. See, for example:
"Cooke, Sir Anthony . . . only son of John Coke or Cooke (1485–1516) of Gidea Hall, Essex . . . and Alice Saunders (d. 1510), daughter of William Saunders of Banbury, Oxfordshire. He was the great-grandson of Sir Thomas Cook. . . lord mayor of London in 1462–3." [ODNB]