Publish that US writer (6)
I believe the answer is:
runyon
'us writer' is the definition.
(short story writer Damon Runyon)
'publish that' is the wordplay.
'publish' becomes 'run' (running is a kind of publishing).
'that' becomes 'yon' (archaic term meaning 'that').
'run'+'yon'='RUNYON'
(Other definitions for runyon that I've seen before include "Damon, writer 1884-1946" , "Damon --, US author" , "Damon --, US novelist" , "American writer" , "Damon ..., author of Guys and Dolls" .)