The most common and most likely answer to this crossword clue is the 4 letter word TEES
I've seen this crossword clue in The Times
T
E
E
S
More clues leading to the result TEES
Buddy explains!

The straight part
We know that "river" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case river = tees.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does TEES mean?
TEES noun- The name of the Latin-script letter T.
- Something shaped like the letter T.
- (clothing) T-shirt.
- A flat area of ground from which players hit their first shots on a golf hole.
- A usually wooden or plastic peg from which a ball is hit.
- The target area of a curling rink
- The mark at which players aim in quoits.
- A finial resembling an umbrella, crowning a dagoba in Indochinese countries.