The question I would ask is simply, why is he doing this. Ten years old is not young, and certainly old enough to be able to understand why unlimited screen time is a problem; it also is old enough to evaluate risks and rewards, as you have found out first hand.
How have you approached this in the past? Is limited screentime simply a rule that he must follow? Or have you discussed with him the whats and the whys of the limit, and approached it as a learning experience?
Remember that in only eight years, he may well be on his own (at least, far more than he is now), and the focus at this age should primarily be helping him to learn to make these decisions correctly for himself. Of course, a ten year old does not have the impulse control of an adult (nor does an 18 year old have the control of a 30 year old, of course, but far more than that of a ten year old); hence why there do need to be actual limits. But, there should be a understanding between you and he as to why those are there. Hopefully that understanding is more than simply having to write an essay (which personally I don't find an effective learning tool here, from my experience, but others do).
Whenever children break rules, I also consider what their motivations are for doing so, and whether they broke the rules because they weren't good rules. I won't tell you how much screen time is appropriate - an hour is less than we allow our nine year old, but it's not unreasonbly less; but I will suggest that you both think yourself, and talk with your child, about whether they are the right limits.
We upped our limits during the pandemic, because we found that our children didn't have any socialization time without their screens; with their screens they can have chats with their friends and play together. It's not as good as playing in person, but it was necessary in our case to have any sort of socialization.
The only way you can find out, though, is to talk to him and find out what his reasoning is. Consider whether you'd be willing to let him have more screen time in exchange for more responsibility - good grades = more time, chores done = more time, that sort of thing. If it's really important to him to have that time, he may well be willing to do other, useful things to earn that responsibility - and then he will feel that he has a way to get what he wants without resorting to lying.
Finally, despite all of that, if you think that your limits are correct, and have done your best to help him learn why they're there - then you should enforce them, and you should take steps necessary. Just think about what the effective steps are, as opposed to punitive.
My son broke a rule related to playing multiplayer games with strangers, which we don't permit; we decided that was something we would stand firm on, and explained to him why. Then we set a restriction on the game in parental controls, that requires him to ask us for permission prior to playing it. It's a trivial thing, but it reminds him that we're thinking about it, and lets us know to pay a bit of attention to it.
Consider doing the same with your son. Put restrictions on the device that require you to approve time. Use appropriate technological safeguards so that the barrier to cheating is higher. It's still likely that a sufficiently motivated child could defeat the safeguards; but you'll probably figure that out (if my son suddenly stopped asking me for time in his game, I'd wonder why), and the point is to have a minimal barrier to stop casual cheating, like the alarms at store exits. Professional thieves don't care about them at all, but it deters casual theft, which (as we're taught in Loss Prevention classes in retain management) is sufficient to deter most people that could be deterred.
