Primate supporting doctor's hanging (5)
I believe the answer is:
drape
'hanging' is the definition.
(thesaurus)
'primate supporting doctor's' is the wordplay.
'primate' becomes 'ape' (I've seen this before).
'supporting' means one lot of letters go next to another (in a down clue, some letters go below others).
'doctor' becomes 'dr' ('Dr' can be a synonym of 'doctor').
'ape' after 'dr' is 'DRAPE'.
(Other definitions for drape that I've seen before include "Hang (cloth) loosely" , "Hang - curtain" , "Pared (anag.)" , "Cover loosely for padre" , "Screen" .)