Pimp’s embarrassed to sleep over (6)
I believe the answer is:
pander
'pimp's' is the definition.
(old-fashioned term for a pimp)
'embarrassed to sleep over' is the wordplay.
'embarrassed' becomes 'red' (someone may blush or go red when embarrassed).
'to sleep' becomes 'nap' (a nap is a short sleep).
'over' says the letters should be written backwards.
'red'+'nap'='rednap'
'rednap' reversed gives 'PANDER'.
(Other definitions for pander that I've seen before include "Cater rto debased tastes" , "Satisfier of someone's baser wishes" , "Gratify weakness or base tastes" , "Cater for debased taste" , "Cater to lowest tastes" .)