Publish that American author (6)
I believe the answer is:
runyon
'american author' is the definition.
(I've seen this before)
'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 --, US novelist" , "Damon ..., writer known for his Broadway stories" , "Damon -, US writer (Guys and Dolls)" , "American author" , "Damon, writer 1884-1946" .)