Reportedly rush into study (4)
I believe the answer is:
read
'study' is the definition.
(I know that study can be written as read)
'reportedly rush' is the wordplay.
'reportedly' indicates a 'sounds like' (homophone) clue.
'rush' becomes 'reed' (I've seen this in another clue).
'reed' sounds like 'READ'.
'into' acts as a link.
(Other definitions for read that I've seen before include "Check, emend (proof)" , "Perused" , "Comprehend written words" , "Stomach" , "Use a book" .)