His mate lays one decrepit redhead off (4)
I believe the answer is:
cock
'his mate lays' is the definition.
(a cock's mate is a hen, a layer of eggs)
'one decrepit redhead off' is the wordplay.
'one decrepit' becomes 'crock' (informal term for a very old person).
'redhead' becomes 'r' (the head/first letter of 'red').
'off' indicates named letters should be taken away.
'crock' with 'r' taken out is 'COCK'.
(Other definitions for cock that I've seen before include "Rooster" , "early bird" , "bull to > raise for shooting" , "Small heap of hay" , "Lift" .)