Insult disheartened lady a little (8)
I believe the answer is:
slightly
'a little' is the definition.
(I've seen this before)
'insult disheartened lady' is the wordplay.
'insult' becomes 'slight' (I've seen this before).
'disheartened' suggests removing the centre.
'lady' with its middle taken out is 'ly'.
'slight'+'ly'='SLIGHTLY'
(Other definitions for slightly that I've seen before include "To a small extent" , "Adverb meaning slender or slim" , "little bit" , "Somewhat" , "Mildly" .)