Society dramatist welcoming second unmarried woman (8)
I believe the answer is:
spinster
'unmarried woman' is the definition.
(term for an older, unmarried woman)
'society dramatist welcoming second' is the wordplay.
'society' becomes 's' (common abbreviation in names of organisations).
'dramatist' becomes 'pinter' (Harold Pinter).
'welcoming' is an insertion indicator (in sense of 'inviting in').
'second' becomes 's' (common abbreviation).
'pinter' going around 's' is 'pinster'.
's'+'pinster'='SPINSTER'
(Other definitions for spinster that I've seen before include "She never married" , "old maid, perhaps" , "woman like Miss Havisham" , "One spurning union activity" , "Elderly unmarried woman" .)