Despair, on reflection, engulfs part of church (4)
I believe the answer is:
apse
'part of church' is the definition.
('apse' is part of 'church')
'despair on reflection engulfs' is the wordplay.
'engulfs' says the answer is hidden in the clue.
'on reflection' says the letters should be written in reverse.
'espa' can be found hidden inside 'despair'.
'espa' in reverse letter order is 'APSE'.
(Other definitions for apse that I've seen before include "Where the altar is usually found in a church" , "Church recess with domed or arched roof" , "Recess at east end of church" , "Church area" , "part of cathedral" .)