Man having rest after tea (7)
I believe the answer is:
charlie
'man' is the definition.
'rest after tea' is the wordplay.
'rest' becomes 'lie' (lying is a kind of resting).
'after' is a charade indicator (letters next to each other).
'tea' becomes 'char' (I've seen this before).
'lie' after 'char' is 'CHARLIE'.
'having' is the link.
(Other definitions for charlie that I've seen before include "tail-ender" , "Comms code C" , "Silly person" , "-- Chaplin" , "succeeds. Bravo!" .)