Farewell promotion Independent editor initially refused essentially (5)
I believe the answer is:
adieu
'farewell' is the definition.
(adieu is a kind of farewell)
'promotion independent editor initially refused essentially' is the wordplay.
'promotion' becomes 'ad' (short for advert).
'initially' indicates taking the first letters.
'essentially' says to take the centre.
The central letter of 'refused' is 'u'.
The first letters of 'independent editor' is 'ie'.
'ad'+'ie'+'u'='ADIEU'
(Other definitions for adieu that I've seen before include "Gallic goodbye" , "Goodbye to the French" , "cheers" , "Vale" , "Ciao, in French" .)