This writer's needing to demonstrate progress (7)
I believe the answer is:
improve
'progress' is the definition.
(thesaurus)
'this writer's needing to demonstrate' is the wordplay.
'this writer's' becomes 'I'm' ('I am' referring to the writer of the clue).
'needing' is a charade indicator (letters next to each other).
'to demonstrate' becomes 'prove' (synonyms).
'im'+'prove'='IMPROVE'
(Other definitions for improve that I've seen before include "Enhance" , "Polish" , "Make better in quality" , "Make progress" , "Change for the better" .)