Had hopes to beat despair (7)
I believe the answer is:
aspired
'had hopes' is the definition.
(I've seen this in another clue)
'beat despair' is the wordplay.
'beat' indicates anagramming the letters.
'despair' with letters rearranged gives 'ASPIRED'.
'to' acts as a link.
(Other definitions for aspired that I've seen before include "Was most anxious to" , "Had ambition" , "Had an ambitious plan or lofty goal" , "Aimed" , "Hoped" .)