This Henry that you are asking about was the son of William Gascoigne and an Elizabeth.Henry had a brother, Robert.They appear to be the only children of William.
Henry married an Elizabeth.I have seen where she was called a Fisher, but have not seen the documentation.Henry left a Will naming wife, Elizabeth, and children--Henry, William, Thomas, John, Elizabeth.
I am interested in William in that I am trying to find out when he arrived in Accomack Co. VA.
Gascoigne is a well-known family in England.However, the renaming compounds things and not all of the branches have been documented.
Accomack Co. VA has a fairly good selection of books abstracted from the courthouse records, etc. if your library has them.I would suggest you take a look as they may answer some of your questions and give you some clues.