The most common and most likely answer to this crossword clue is the 13 plus 4 letter word EGG-AND-SPOON RACE
I've seen this crossword clue in Independent
E
G
G
-
A
N
D
-
S
P
O
O
N
R
A
C
E
Buddy explains!
The straight part
We know that "school competition" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case school competition = egg-and-spoon race.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.