Dear Pat, I have a goldmine of information for you and there is lots more out there. Luckily, you are part of a well-researched family. I'll give you most of what I have, and then you can email me for more if you'd like or check out some sources for yourself. I don't have specific dates for your Henry, but I have plenty of other ones.
I'll begin with the Cudworths in America (even though I have info back to the 1350's in Eng.)
James Cudworth b. 1604 Allerston, N. Yorkshire, England d. Nov. 1682 in London, London, England m. 1. Mary Sarah Goodman 2. Sarah Mary Stoughton 3. Mrs. Mary Cudworth children: James II (unfortunately, I can't find his dob at the moment) and others *note-James I was a Lieut. Gov in Scituate,MA
James Cudworth II m. Mary Howland (whose uncle was James Howland who came over on the Mayflower in 1620) Children: James Cudworth III and others
James Cudworth III m. Betty Hatch Children: David and others
David Cudworth b. 1695 m. Abigail Josslin (or Jocelyn) Children: Lydia, Charles, and others
Charles Cudworth m. 1.May Drinkwater (descended from George Soule of the Mayflower) *note-the info I have says that 5 Cudworth boys married 5 of the Drinkwater girls, kinda interesting 2. Content Ramsdell Children with 1.:Josslin and several others
Josslin Cudworth b. 1774 m. Sarah Reed Children: John, Nathaniel, Leavitt, and Sarah
John Cudworth b. 1806 m. Amy Spaulding about 1837 Children: Sarah Jane(1838), Marcia (1839), Hattie (1842), John Wesley (1844), Henry Nathaniel, and Boardman Leavitt (1846)
I have spouses and even some children for all of these children. I am descended from Boardman Cudworth, through his son Harold, who was my g-grandfather. A great resource that I've never gotten to check out but that comes highly recommended is Records of the Cudworth Family by Arthur G. Cudworth. It contains info from at least 1100 A.D. Also, you can look at info on Ralph Cudworth, who was a famous author, philosopher, and theologian. He is listed in several online encyclopedias. There is even a Cudworth house in Scituate, MA that serves as a sort of museum (James I was friends with men like John Alden and Miles Standish). I hope that this information helps you. If you need any more info, don't hesitate to write me at firstname.lastname@example.org. Good luck!