Live with rest, not posh (6)
I believe the answer is:
reside
'live' is the definition.
(I know that reside is a type of live)
'rest not posh' is the wordplay.
'rest' becomes 'residue' (residue can mean the remains or rest of something).
'not' suggests deleting specific letters.
'posh' becomes 'u' (abbreviation for upper-class - opposite to non-U).
'residue' with 'u' taken away is 'RESIDE'.
'with' is the link.
(Other definitions for reside that I've seen before include "Occupy a place" , "Stay - desire (anag)" , "Live (at an address)" , "Make one's home permanently" , "Dwell in desire" .)