Put off detective writer, but not getting cross (5)
I believe the answer is:
deter
'put off' is the definition.
(thesaurus)
'detective writer but not getting cross' is the wordplay.
'detective writer' becomes 'dexter' (Colin Dexter).
'but not getting' suggests deleting specific letters.
'cross' becomes 'x'.
'dexter' with 'x' removed is 'DETER'.
(Other definitions for deter that I've seen before include "Put (someone) off" , "Check" , "Discourage, put off" , "Cause not to do something" , "Discourage (an activity)" .)