To anyone out there interested in the Geappen/Gapen/Gapin et al famiy from Eling.
I have now visited the Winchester Records Office myself and have seen the microfiche which contains the "photocopy" of the actual baptism record for John Gapen.
Whoever his mother was, she was definitely NOT named Anne.
Her name is 4 letters long, and the last 2 letters are certainly "ne"
The print is very fuzzy, but all 9 baptism entries on that page have been handwritten by someone named Ashe, Ashen or Ashes, who has a very distinctive way of writing his A's. Hence I know the first letter of the mother's name is not an A. I suspect it is actually a capital J (ie the looped sort), making the mother's name Jane. As the only William Geappen I can find living in Eling at the time was married to Jane Green, I am happy in my own mind that John is the son of William and Jane.