King enters rising river for sport (5)
I believe the answer is:
skeet
'sport' is the definition.
(thesaurus)
'king enters rising river' is the wordplay.
'king' becomes 'K' (chess abbreviation).
'enters' is an insertion indicator.
'rising' says the letters should be written in reverse (letters written upwards in a down clue).
'river' becomes 'tees' (River Tees ).
'tees' back-to-front is 'seet'.
'k' put within 'seet' is 'SKEET'.
'for' is the link.
(Other definitions for skeet that I've seen before include "US version of a shooting sport involving clay pigeons" , "American clay pigeon shooting" , "Form of clay pigeon shooting" , "A form of clay-pigeon shooting" .)