Not quite willing to study (4)
I believe the answer is:
read
'study' is the definition.
('read' can be a synonym of 'study')
'not quite willing' is the wordplay.
'not quite' means to remove the last letter.
'willing' becomes 'ready' ('ready' is associated in meaning with 'willing').
'ready' with its last letter taken off is 'READ'.
'to' acts as a link.
(Other definitions for read that I've seen before include "Use a book" , "Undertake a course of study" , "what follows 18 [OMASUM] in a 17 [RUMINANT]" , "Study (for a degree?)" , "See - peruse - understand" .)