Irritable agent going without sleep (6)
I believe the answer is:
snappy
'irritable' is the definition.
(I know that irritable can be written as snappy)
'agent going without sleep' is the wordplay.
'agent' becomes 'spy' (spy is a kind of agent).
'going without' is an insertion indicator ('without' can mean 'outside').
'sleep' becomes 'nap' (a nap is a short sleep).
'spy' enclosing 'nap' is 'SNAPPY'.
(Other definitions for snappy that I've seen before include "Smart - lively" , "sort of dresser" , "(Make it) quick" , "Bad-tempered" , "Quick and energetic" .)