Thank you for the clarification. I am also reminded that I forgot to include the link to the Google Books preview of AR: https://books.google.com/books?id=XLqEWwa7fT8C&pg=PA211&dq=Thomas+heigham+catherine+cotton&hl=en&sa=X&ved=0ahUKEwjz9PuTu9DNAhWK1B4KHcRoC7EQ6AEIHDAA#v=onepage&q=Thomas%20heigham%20catherine%20cotton&f=false J.B. Threlfall discussed the Cottons, including Thomas Heigham and Margaret Cotton, in his 1970 "The Ancestry of My Children" and then in greater detail in his 1988 "The Ancestry of Thomas Bradbury (1611-1695) and His Wife Mary (Perkins) Bradbury (of Salisbury" (again, I have no access to these).

If the Bures line is correct there might be the following: Henry I, King of England (illegit.) Robert "of Caen," Earl of Gloucester = Mabel FitzHamon Maud of Gloucester = Ranulph, 2nd Earl of Chester Hugh, 3rd Earl of Chester = Bertrada de Montfort Agnes of Chester = William de Ferrers, 4th Earl of Derby William de Ferrers, 5th Earl of Derby = Sibyl Marshal Joan de Ferrers = Sir Robert Aguillon Isabel Aguillon = Hugh, Lord Bardolf Margery Bardolf = Sir Michael Poynings Thomas, Baron Poynings = Agnes de Rokesle Michael, Baron Poynings = Joan ___, widow of Sir John Molyns Alice Poynings = Sir Andrew Bures Andrew Bures = Alice Spencer Andrew Bures = Catherine Shiltenne William Bures = Elizabeth Rowse William Bures = Joan ____ (? Markham) Robert Bures = Joan Bucke [daughter] Bures = Robert or Thomas King (Or William King?) Judith King = Bishop Aylmer