The most common and most likely answer to this crossword clue is the 3 plus 5 letter word DOG TRACK
I've seen this crossword clue in Financial Times
D
O
G
T
R
A
C
K
Buddy explains!
The straight part
We know that "where the greyhounds run" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case where the greyhounds run = dog track.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.