What health inspector seeks in hospital, eyeing rubbish (7)
I believe the answer is:
hygiene
'what health inspector seeks' is the definition.
'hospital eyeing rubbish' is the wordplay.
'hospital' becomes 'h'.
'rubbish' indicates anagramming the letters.
'eyeing' anagrammed gives 'ygiene'.
'h'+'ygiene'='HYGIENE'
'in' is the link.
(Other definitions for hygiene that I've seen before include "Cleanliness aimed at maintaining health" , "Cleanliness, sanitation" , "Clean and healthy practices" , "Sanitary practices conducive to health" , "Personal cleanliness" .)