Connection between Ainsworth and Mayflower?

Does anyone know of a connection between the Ainsworth's of Claremont, NH, and someone who came off from the Mayflower?

My father always told how his mother, my grandmother, Carolyn Woodward Maiden Name Ainsworth was able to write out her familiy tree from memory. She always said that my dad was related to someone who came off from the Mayflower though my dad believes it was through the Ainsworth side. My dad does not remeber how he was related since he wasn't interested in geneology at the time. After my grandmother died we did find some papers that were written out but they only followed the geneology of the Ainsworth's through the male lineage.

I already know that I am a distant relation to John Howland through my mother's side since my Aunt, and other grandmother have done research there the past number of years through my other grandmother Lorraine Lambert maiden name Woodard, and realize woodward/ woodard are same familiy so Iam probably related to myself lol.