A depression overcoming that woman’s supporter (8)
I believe the answer is:
adherent
'supporter' is the definition.
(I've seen this in another clue)
'a depression overcoming that woman's' is the wordplay.
'depression' becomes 'dent' (synonyms).
'overcoming' is an insertion indicator.
'that woman' becomes 'her' ('her' refers to a woman).
'dent' placed around 'her' is 'dherent'.
'a'+'dherent'='ADHERENT'
(Other definitions for adherent that I've seen before include "One supports" , "Supporter, follower" , "Loyal follower - sticks close" , "Disciple, follower" , "Weep" .)