Background of short story by playwright finally (7)
I believe the answer is:
context
'background' is the definition.
(thesaurus)
'short story by playwright finally' is the wordplay.
'short story' becomes 'conte' (I've seen this in another clue).
'by' becomes 'x' (as in 'multiplied by').
'finally' says to take the final letters.
The final letter of 'playwright' is 't'.
'conte'+'x'+'t'='CONTEXT'
'of' acts as a link.
(Other definitions for context that I've seen before include "Setting of discourse" , "what they relate to" , "Frame of reference" , "Surrounding setting" , "Words surrounding a word or phrase that may fix its true meaning" .)