Centre for cultural development where favourite food captivates Irish going over (5,4)
I believe the answer is:
petri dish
'centre for cultural development' is the definition.
(a petri dish is used to culture microorganisms)
'favourite food captivates irish going over' is the wordplay.
'favourite' becomes 'pet' ('pet' can be a synonym of 'favourite').
'food' becomes 'dish' (dish is a kind of food).
'captivates' is an insertion indicator.
'irish' becomes 'ir' (abbreviation).
'going over' says the letters should be written backwards.
'ir' written backwards gives 'ri'.
'pet'+'dish'='petdish'
'petdish' enclosing 'ri' is 'PETRI DISH'.
'where' is the link.
(Other definitions for petri dish that I've seen before include "Shallow bowl used to culture bacteria -- priest hid (anag)" , "Type of plate used for culturing bacteria etc" , "Shallow plate for lab study" , "Container for growing cultures" .)