Ad lib concert after the writer's put out (9)
I believe the answer is:
impromptu
'ad lib' is the definition.
(synonyms)
'concert after the writer's put out' is the wordplay.
'concert' becomes 'prom' (abbreviation for promenade concert).
'after' means one lot of letters go next to another.
'the writer's' becomes 'im' (both could mean 'the crossword setter is').
'out' is an anagram indicator (out can mean wrong or inaccurate).
'put' anagrammed gives 'ptu'.
'prom' put after 'im' is 'improm'.
'improm'+'ptu'='IMPROMPTU'
(Other definitions for impromptu that I've seen before include "Off-the-cuff" , "Ad-lib" , "Extemporaneous speech" , "Spontaneous" , "Improvised" .)