Cuckoo smeared this with small beak? (12)
I believe the answer is:
headmistress
'beak?' is the definition.
('beak' can refer to a headteacher)
'cuckoo smeared this with small' is the wordplay.
'cuckoo' is an anagram indicator (I've seen this in other clues).
'with' means one lot of letters goes inside another.
'small' becomes 's' (abbreviation - e.g. clothes size).
'smeared'+'this'='smearedthis'
'smearedthis' with letters rearranged gives 'headmitress'.
'headmitress' enclosing 's' is 'HEADMISTRESS'.
(Other definitions for headmistress that I've seen before include "School principal" , "Teacher" , "Person in charge of a school" , "Female school leader" .)