Be left at home with their mess (7)
I believe the answer is:
inherit
'be left' is the definition.
(to inherit something is to be left it)
'home with their mess' is the wordplay.
'home' becomes 'in' ('he's home' can mean 'he's in').
'with' is a charade indicator (letters next to each other).
'mess' indicates anagramming the letters.
'their' anagrammed gives 'herit'.
'in'+'herit'='INHERIT'
'at' acts as a link.
(Other definitions for inherit that I've seen before include "Receive under terms of will" , "Obtain as legacy" , "Receive by succession or via a will" , "Receive (as part of estate)" , "Acquire from ancestors" .)