I found one sentence while I was reading a book:
He did not study hard on the test.
Is this sentence correct in reference to the preposition 'on' ?
I think the better version is:
He did not study hard for the test.
We could say "study on a lesson" or "study on Engish", according to Search in Google Books.
And I think "on" probably means "relating" in this situation. See ON | meaning in the Cambridge English Dictionary:
on preposition (RELATING) B1 relating to:
a book on pregnancy
Her talk is on Italian women's literature.
The minister has refused to comment on the allegations.
Criticism has no effect on him.
Do the police have anything on you (= do they have any information about you that can be used against you)?
"Study hard for" is OK.
If "study on a test" is valid, the closest meaning I can find in the dictionary would be "paying". See ON | meaning in the Cambridge English Dictionary:
on preposition (MONEY)
A2 used to show something for which a payment is made:
He spent €180 on a hat.
I've wasted a lot of money on this car.
We made a big profit on that deal.
How much interest are you paying on the loan?
So "study hard on a test" is probably wrong.