As Doug mentioned, you can build out your ancestors here, citing all the sources that you have/find along the way. While you may not have a Clayton surname ancestor that arrived on the Mayflower, it's still possible that you may eventually find that your "family hearsay" of Mayflower ancestors through your Clayton line is true. You have many ancestors that branch off through the generations of maternal lines of your Clayton ancestry going back that far, and that would have a different surname than Clayton. It's possible that one of those could eventually lead back to a Mayflower passenger.
Good luck and happy hunting!