I haven't got too much on the Yorkshire Gawthorps, as my own Gawthrop ancestors come from Histon in Cambridgeshire, and I haven't yet been able to find any links between the two groups, although I'm sure they must exist.
Have you tried the internet site WWW.Gawthrop.orgThis gives quite a lot of information on the Yorkshire Gawthrop's.There is also a book, which you may well already be familiar with - It traces the descendents of Thomas Gawthrop, a Quaker from Skipton in Yorkshire who emigrated to America.This is I think, available at www.willowbend.net.It is called 'A genealogical record of the descendants in the male line of Thomas Gawthrop'.
I also have copies of two wills from Yorkshire Gawthrops.The first is the will of a Rev. Christopher Gawthrop who died 1787, and had lands in the Sedbergh area of Yorkshire.The second will is that of his wife, Elizabeth Gawthrop, who died 1793.These are both quite lengthy documents, but if you think they may have a connection to your own tree, I am willing to scan them and email them to you.
Other wills available (although I don't currently have copies of them) are: 1541 - Roger Gawkthrop, of Bradley Kildwicke in Yorkshire (In York Registry) 1559 - Richard Gawthrop, of Carlton Craven in Yorkshire (In York registry) 1656 - Gregory Gawthrop, husbandman of Kilton, Worksop, Nottinghamshire. (Registered at PCC)