A cook's endless pain (4)
I believe the answer is:
ache
'pain' is the definition.
(ache is a kind of pain)
'a cook's endless' is the wordplay.
'cook' becomes 'chef' (both can mean someone who cooks).
'endless' means to remove the last letter.
'chef' with its last letter removed is 'che'.
'a'+'che'='ACHE'
(Other definitions for ache that I've seen before include "Hanker" , "yearning" , "agony" , "Hurt; yearn" , "painful" .)