The most common and most likely answer to this crossword clue is the 4 plus 6 letter word LAWN TENNIS
I've seen this crossword clue in The Times
L
A
W
N
T
E
N
N
I
S
Buddy explains!
The straight part
We know that "competition at court" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case competition at court = lawn tennis.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.