Thatcher perhaps and supporter losing head (7)
I believe the answer is:
artisan
'thatcher perhaps' is the definition.
(thatcher is a kind of artisan)
'supporter losing head' is the wordplay.
'supporter' becomes 'partisan' (partisan is a kind of supporter).
'losing head' means to remove the first letter.
'partisan' with its initial letter taken off is 'ARTISAN'.
'and' is the link.
(Other definitions for artisan that I've seen before include "man with skill" , "Skilled tradesman" , "Skilled craftsperson" , "Sinatra (anag) -- skilled manual worker" , "Handy worker" .)