Restaurant serving each man or woman right away (6)
I believe the answer is:
eatery
'restaurant' is the definition.
(place to eat)
'each man or woman right away' is the wordplay.
'each' becomes 'ea' (abbreviation).
'man or woman' becomes 'Terry' (unisex name).
'right' becomes 'r' (common abbreviation).
'away' is a deletion indicator.
'terry' with 'r' taken out is 'tery'.
'ea'+'tery'='EATERY'
'serving' is the link.
(Other definitions for eatery that I've seen before include "diner" , "Place serving food" , "Brasserie" , "[BISTRO]" , "Restaurant, cafe" .)