Conservative connection in prison (5)
I believe the answer is:
Here is my best explanation:
'prison' is the definition.
'conservative connection' is the subsidiary indication.
'conservative' becomes 'c' (abbreviation commonly used in UK politics).
'connection' becomes 'link' (link is a kind of connection**).
'in' is the link.
(Other definitions for clink that I've seen before include "Prison - knock glasses together", "Ringing sound - prison", "jug", "1 ac [COOLER]", "Prison (slang)", "Rattle of coins; prison (slang)", "Sound made when hitting glass" and "Old London prison".)