A comment about City Director's first story (8)
I believe the answer is:
anecdote
'story' is the definition.
(thesaurus)
'a comment about city director's first' is the wordplay.
'comment' becomes 'note' (note is a kind of comment**).
'about' is an insertion indicator.
'city' becomes 'ec' (the City of London postcode).
'first' suggests taking the first letters.
The initial letter of 'directors' is 'd'.
'ec'+'d'='ecd'
'note' enclosing 'ecd' is 'necdote'.
'a'+'necdote'='ANECDOTE'
(Other definitions for anecdote that I've seen before include "Yarn" , "Short amusing tale" , "Reminiscence, story" , "Short entertaining story" , "Brief tale" .)