Recess, beset by rising despair (4)
I believe the answer is:
apse
'recess' is the definition.
(apse is a kind of recess)
'beset by rising despair' is the wordplay.
'rising' shows that the letters should be reversed in order (going up in a down clue).
'beset by' says the answer is hidden in the clue (the answer is surrounded or beset within the clue).
'espa' can be found hidden inside 'despair'.
'espa' back-to-front is 'APSE'.
(Other definitions for apse that I've seen before include "Semicircular end of a church, behind the altar" , "In sacred space" , "where praetor sat" , "room for an altar" , "space for altar" .)