Impress popular writer (8)
I believe the answer is:
inscribe
'impress' is the definition.
(thesaurus)
'popular writer' is the wordplay.
'popular' becomes 'in' (something that's 'in' is fashionable).
'writer' becomes 'scribe' (scribe is a kind of writer).
'in'+'scribe'='INSCRIBE'
(Other definitions for inscribe that I've seen before include "Write a dedication to someone in (a book)" , "Write or carve on something as a lasting record" , "Write, engrave, or print as a lasting record" , "Engrave or write upon" , "autograph" .)