I managed to achieve this through modifying $VIMRUNTIME/syntax/html.vim. Make a copy to ~/.vim/syntax/html.vim (.vim is named vimfiles on Windows), and replace the original syntax definition
syn region htmlLink start="<a\>\_[^>]*\<href\>" end="</a>"me=e-4 contains=@Spell,htmlTag,htmlEndTag,htmlSpecialChar,htmlPreProc,htmlComment,javaScript,@htmlPreproc
with the following:
syn region htmlLink start="<a\>\_[^>]*\<href\>" end="</a>"me=e-4 keepend contains=@Spell,htmlTag,htmlEndTag,htmlSpecialChar,htmlPreProc,htmlComment,htmlLinkText,javaScript,@htmlPreproc
syn match htmlLinkText contained contains=@Spell,htmlTag,htmlEndTag,htmlSpecialChar,htmlPreProc,htmlComment,htmlLinkText,javaScript,@htmlPreproc "^\s*\zs.\{-}\ze\s*$"
syn match htmlLinkText contained contains=@Spell,htmlTag,htmlEndTag,htmlSpecialChar,htmlPreProc,htmlComment,htmlLinkText,javaScript,@htmlPreproc "\S.\{-}\ze\s*$"
Further down, change
HtmlHiLink htmlLink Underlined
to
HtmlHiLink htmlLinkText Underlined
Voila! Basically, this introduces another contained syntax group htmlLinkText, which does not match leading and trailing whitespace, and applies the highlighting to that instead.
