Put back in rider's gear right after eventing debut (8)
I believe the answer is:
reinsert
'put back in' is the definition.
(insert for a second time)
'rider's gear right after eventing debut' is the wordplay.
'rider's gear' becomes 'reins' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'right' becomes 'rt' (abbreviation as in Rt Hon- the Right Honourable).
'after' is a charade indicator (letters next to each other).
'debut' says to take the initial letters.
The first letter of 'eventing' is 'e'.
'rt' put after 'e' is 'ert'.
'reins'+'ert'='REINSERT'
Can you help me to learn more?
(Other definitions for reinsert that I've seen before include "Enter, sir! (anag) - put in again" , "plant back" , "Put back in" .)