Remains in an institution (4,4)
I believe the answer is:
rest home
'an institution' is the definition.
(I know that rest home is a type of social welfare institution)
'remains in' is the wordplay.
'remains' becomes 'rest' (both can mean what's left over).
'in' becomes 'home' ('he's in' can mean 'he's home').
'rest'+'home'='REST-HOME'
(Other definitions for rest home that I've seen before include "place to stay" , "residence" , "old institution" , "Place for the old" , "place for the frail elderly" .)