Choke, having to dine with the fourth estate (8)
I believe the answer is:
suppress
'choke' is the definition.
(I know that choke is a more specific form of the action dampen)
'to dine with the fourth estate' is the wordplay.
'to dine' becomes 'sup' (synonyms).
'with' means one lot of letters go next to another.
'the fourth estate' becomes 'press' (term for the press).
'sup'+'press'='SUPPRESS'
'having' acts as a link.
(Other definitions for suppress that I've seen before include "Prevent (from being published)" , "Forcibly keep down" , "Crush, put down" , "Prevent from being made public" , "Prevent, inhibit" .)