The writer backed the writer’s award (4)
I believe the answer is:
emmy
'award' is the definition.
(I've seen this before)
'the writer backed the writer's' is the wordplay.
'the writer' becomes 'me' (the person who wrote the clues).
'backed' is a reversal indicator.
'the writer's' becomes 'my' (belonging to the person who set the crossword).
'me' back-to-front is 'em'.
'em'+'my'='EMMY'
(Other definitions for emmy that I've seen before include "Least complicated" , "Uncle Sam's award" , "British television award" , "TV award; girl's name" , "Little girl" .)