Assigned daughter to follow a writer (8)
I believe the answer is:
ascribed
'assigned' is the definition.
('ascribe' can be a synonym of 'assign')
'daughter to follow a writer' is the wordplay.
'daughter' becomes 'd' (genealogical abbreviation).
'to follow' says to put letters next to each other (some letters go after or follow others).
'writer' becomes 'scribe' (scribe is a kind of writer).
'a'+'scribe'='ascribe'
'd' put after 'ascribe' is 'ASCRIBED'.
(Other definitions for ascribed that I've seen before include "Put down" , "Given the attribution" , "Attributed, regarded as originated by" .)