In favour of sister taking on unfinished part of speech (7)
I believe the answer is:
pronoun
'part of speech' is the definition.
(pronouns are a part of speech)
'in favour of sister taking on unfinished' is the wordplay.
'in favour of' becomes 'pro' (being pro a particular cause).
'sister' becomes 'nun' (sister is a term of address for a nun).
'taking' is an insertion indicator.
'unfinished' means to remove the last letter.
'on' with its last letter taken off is 'o'.
'nun' enclosing 'o' is 'noun'.
'pro'+'noun'='PRONOUN'
(Other definitions for pronoun that I've seen before include "Category of word" , "He or she etc." , "Possibly he" , ""She" or "him", for example" , "Part of speech" .)