Xmas missive inspiring old western playwright (4,6)
I believe the answer is:
noel coward
'playwright' is the definition.
(I know that Noel Coward is a playwright)
'xmas missive inspiring old western' is the wordplay.
'xmas' becomes 'noel' (I've seen this before).
'missive' becomes 'card' (missive can mean a written communication).
'inspiring' is an insertion indicator.
'old' becomes 'o' (abbreviation).
'western' becomes 'w'.
'o'+'w'='ow'
'card' placed around 'ow' is 'coward'.
'noel'+'coward'='NOEL COWARD'
(Other definitions for noel coward that I've seen before include "Old playwright" , "dramatist" , "He wrote" .)