See Paul Reed's article in vol. 9 (1988) of "The Genealogist" [TG] on
the origins of the Harleston family of South Carolina. He itemizes
the children of John Harleston and Alice Clopton and does not include
the Admiral among them. He goes further to say that there is no
apparent connection between the Admiral and this family. The ODNB bio
of the Admiral specifies no parents for him - presumably they're
unknown.
I assume the "well known website" that claims that ""His family tree
has been well researched by
specialists in medieval genealogy" is this one:
http://www.guernsey-society.org.uk/donkipedia/index.php5?title=Sir_Richard_Harliston
Note that it does not identify the "specialists in medieval genealogy"
who supposedly researched his ancestry - hardly a reliable site. And
the descent that it gives from King John has already been shown to be
wrong in at least two places - notably including the identification of
Richard's parents.
There's probably a good reason why you "have been unable to find
anything that support[s] this statement": - because there IS nothing
to support it. Just because a genealogy is on the web doesn't means
it has any reliability. If you really want to pursue this, start with
the ODNB bio and Paul Reed's article, both of which have already been
cited here, and follow the sources they list.
Paul Reed's article is almost certainly not on-line - it's far too
recent. You'll probably have to look for the periodical in a library
(or ask your library to obtain a copy via inter-library loan if
possible). ODNB may be available to you on-line if your local library
has subscribed to its service. Or you could look for the old edition
of the DNB which many libraries have in book form.
The DNB is online.
I have a book about the early life of Margaret Harleston (Titled the
Fortress) and her best friend Jacqueline Wallis daughter of Geoffry Wallis a
native of the Channel Islands. Some of the story is based at Richard
Harleston home of Oakland Park in Portsmouth. On four occasions an Aunty of
Margaret is mentioned that being the Lady Fitzhugh. Does anyone have any
idea on who the Lady Fitzhugh might be? This book also states that Geoffry
Wallis was Sir Philippe de Carteret's brother in law and that Jacqueline was
his niece. It appears Sir Philippe de Carteret husband of Margaret had been
previously married. His wife had died leaving a son Edouard de Carteret
prior to his marrage to Margaret Harleston.