Minister starts to visit Irish churches at random (5)
I believe the answer is:
vicar
'minister' is the definition.
(a religious minister)
'starts to visit irish churches at random' is the wordplay.
'starts to' says to take the initial letters.
The first letters of 'visit irish churches at random' is 'VICAR'.
(Other definitions for vicar that I've seen before include "Serviceman" , "Minister" , "Local priest" , "Parson" , "Man with a dog collar" .)