Mature writer after Irish backing (5)
I believe the answer is:
ripen
'mature' is the definition.
(ripening is a kind of maturing)
'writer after irish backing' is the wordplay.
'writer' becomes 'pen' (I've seen this before).
'after' is a charade indicator (letters next to each other).
'irish' becomes 'ir' (abbreviation).
'backing' shows that the letters should be reversed in order.
'ir' backwards is 'ri'.
'pen' after 'ri' is 'RIPEN'.
(Other definitions for ripen that I've seen before include "Become mature" , "Become ready for harvesting" , "Season" , "Grow to maturity like fruit" , "Become ready to pick" .)