Supporter at home was hard to indoctrinate (9)
I believe the answer is:
'indoctrinate' is the definition.
(brainwashing is a kind of indoctrinating)
'supporter at home was hard' is the wordplay.
'supporter' becomes 'bra' (I've seen this before).
'at home' becomes 'in' ('in' can be similar in meaning to 'at home').
'hard' becomes 'h'.
'bra'+'in'+'was'+'h'='BRAINWASH'
'to' is the link.
(Other definitions for brainwash that I've seen before include "clean mind?" , "Subject to systematic indoctrination" , "Apply pressure" , "Completely indoctrinate" , "Subject someone to stressful indoctrination" .)