The most common and most likely answer to this crossword clue is the 10 letter word INVIGILATE
I've seen this crossword clue in Independent
I
N
V
I
G
I
L
A
T
E
More clues leading to the result INVIGILATE
Buddy explains!

The straight part
We know that "to supervise kids (who are sitting exams)" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case to supervise kids (who are sitting exams) = invigilate.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 INVIGILATE mean?
INVIGILATE verb- To oversee a test or exam.