This writer's continuing with the unfinished story (4)
I believe the answer is:
myth
'story' is the definition.
(myth is a kind of story)
'this writer's continuing with the unfinished' is the wordplay.
'this writer's' becomes 'my' (belonging to the person who set the crossword).
'continuing with' means one lot of letters go next to another.
'unfinished' means to remove the last letter.
'the' with its final letter taken away is 'th'.
'my'+'th'='MYTH'
(Other definitions for myth that I've seen before include "Tall tale" , "lie!" , "A US president; a UK PM" , "Traditional allegory" , "Legend; fable" .)