Dusting sign on top of library during cleaning period? (10)
I believe the answer is:
sprinkling
'dusting' is the definition.
(sprinkling is a kind of dusting)
'sign on top of library during cleaning period?' is the wordplay.
'sign' becomes 'ink' (ink can mean to sign a contract).
'on top of' suggests taking the first letters.
'during' indicates putting letters inside.
'cleaning period?' becomes 'spring' (as in spring cleaning).
The initial letter of 'library' is 'l'.
'ink'+'l'='inkl'
'inkl' put into 'spring' is 'SPRINKLING'.
(Other definitions for sprinkling that I've seen before include "Drizzling" , "Not much water" , "trickle" .)