His out-tray, not small, could be specialist (9)
I believe the answer is:
authority
'specialist' is the definition.
(thesaurus)
'his out-tray not small could be' is the wordplay.
'not' is a deletion indicator.
'small' becomes 's' (abbreviation - e.g. clothes size).
'could be' is an anagram indicator.
'his'+'out'+'tray'='hisouttray'
'hisouttray' with 's' taken away is 'hiouttray'.
'hiouttray' anagrammed gives 'AUTHORITY'.
(Other definitions for authority that I've seen before include "Power, influence" , "Power to expercise control" , "Mandate" , "Expert; permission" .)