Captain Cooks family tree. From what I read only one of his sisters line produce children who went on to produce more children and that was Margaret COOK b:1742 who married George FLECK. Now interestingly I noted you mentioned the surname CARTER - one of Margaret COOK and George FLECK's daughters Grace FLECK b: 1770 (who would have been Cook's niece) married a John CARTER and they had a number of daughters if you could find one that possibly married a SAUNDERS then you may find your connection.
But like all things it maybe just a family myth.
Below is a link to Captain Cook's family tree.http://www.captcook-ne.co.uk/ccne/familytree.htm