Burnt-out cases end up here, lost around hospital (7)
I believe the answer is:
ashtray
'burnt-out cases end up here' is the definition.
(as in cigarette cases)
'lost around hospital' is the wordplay.
'lost' becomes 'astray' (I've seen this before).
'around' indicates putting letters inside.
'hospital' becomes 'h'.
'astray' placed around 'h' is 'ASHTRAY'.
(Other definitions for ashtray that I've seen before include "Smelly receptacle" , "Where to rest cigarette" , "where smoking remains are found" , "Smoker's receptacle" , "Smoker's aid" .)