Aimed to resolve despair (7)
I believe the answer is:
aspired
'aimed' is the definition.
('aspire' can be a synonym of 'aim')
'to resolve despair' is the wordplay.
'to resolve' is an anagram indicator (I've seen 'resolving' mean this).
'despair' is an anagram of 'ASPIRED'.
(Other definitions for aspired that I've seen before include "Aimed" , "Hoped" , "Was most anxious to" , "Had an ambitious plan or lofty goal" , "Had ambition" .)