Drink provided for detectives with hesitation (5)
I believe the answer is:
cider
'drink' is the definition.
(cider is a kind of drink)
'detectives with hesitation' is the wordplay.
'detectives' becomes 'CID' (abbreviation for Criminal Investigation Department).
'with' is a charade indicator (letters next to each other).
'hesitation' becomes 'er' (sound made while hesitating - um, er).
'cid'+'er'='CIDER'
'provided for' acts as a link.
(Other definitions for cider that I've seen before include "Apple-based beverage" , "Alcoholic drink made from apples" , "Apple-based drink" , "Drink made from pears" , "Beverage from apple juice" .)