Authority to keep daughter’s cosmetic (6)
I believe the answer is:
powder
'cosmetic' is the definition.
(I have seen 'Facial cosmetic ' mean 'powder' so perhaps 'cosmetic' could also mean 'powder')
'authority to keep daughter's' is the wordplay.
'authority' becomes 'power' (I've seen this in another clue).
'to keep' means one lot of letters goes inside another.
'daughter' becomes 'd' (genealogical abbreviation).
'power' enclosing 'd' is 'POWDER'.
(Other definitions for powder that I've seen before include "secured by a magazine" , "Facial cosmetic" , "Medical preparation" , "Small loose particles" , "Beauty aid" .)