There at least two Lufkin families living in Harwich, Essex about 1550 to 1625 or so. John Lufkin, Brian or Briant Lufkin and:
Jone (Joan) Lufkin who married William Oliff alias Lambert 21 January 1578/79.
Perhaps your Lufkins originally came from Harwich? You can access the original St. Nicholas parish registers and other information at: seax.essexcc.gov.uk