Allow the hospital to treat Gary for lassitude (8)
I believe the answer is:
lethargy
'lassitude' is the definition.
(synonyms)
'allow the hospital to treat gary' is the wordplay.
'allow' becomes 'le' (I've seen this in other clues).
'the' becomes 't' (the is pronounced as a 't' sound in some dialects).
'hospital' becomes 'H' (abbreviation for hospital).
'to treat' is an anagram indicator.
'gary' with letters rearranged gives 'argy'.
'le'+'t'+'h'+'argy'='LETHARGY'
'for' acts as a link.
(Other definitions for lethargy that I've seen before include "Unnatural drowsiness" , "'Lassitude, langour (8)'" , "Lack of energy and enthusiasm" , "Extreme lack of energy, listlessness" , "Lassitude, languor" .)