Mystery writer not starting with bonus puzzle (7)
I believe the answer is:
nonplus
'puzzle' is the definition.
(synonyms)
'mystery writer not starting with bonus' is the wordplay.
'mystery writer' becomes 'anon' (I've seen this before).
'not starting' means to remove the first letter.
'with' is a charade indicator (letters next to each other).
'bonus' becomes 'plus' (I've seen this before).
'anon' with its initial letter taken away is 'non'.
'non'+'plus'='NONPLUS'
(Other definitions for nonplus that I've seen before include "Perplex completely with negative and positive" , "Floor" , "Flummox" , "Throw into confusion" , "nothing if not negative?" .)