Promote supporter, having an opening (9)
I believe the answer is:
propagate
'promote' is the definition.
(thesaurus)
'supporter having an opening' is the wordplay.
'supporter' becomes 'prop' (I've seen this before).
'having' means one lot of letters go next to another.
'an' becomes 'a'.
'opening' becomes 'gate' (I have seen 'Movable opening ' mean 'gate' so perhaps 'opening' could also mean 'gate'**).
'prop'+'a'+'gate'='PROPAGATE'
(Other definitions for propagate that I've seen before include "Breed from parent stock" , "Multiply, transmit" , "broadcast" , "Disseminate" , "Generate by grafting or layering" .)