I need to write sentences using specific phrasal verbs. I need to knwo if they are possible and supply the equivalent in formal English.

I havent' found the meaning of "let on to" Does it exist?
Thank you

1) One boy had let off a firework in class. Many fireworks were let off on the saint patron’s day.
2) The boy was let off with a warning because he told the truth. The teacher let them off their homework. You’ve worked hard all week. I’ll let you off today.
Let on to:
3) I’ve told you a secret: promise you won’t let on
It was very hot and I took my sweater off.
4) John, take off your coat, come through and meet the others.
Dad took the day off to come with me.
%5)They took twenty per cent off the list price.
The plane took off after refueling.
She became jealous when his career started taking off.
6) Don’t take too much work on – the extra cash isn’t worth it. She has taken on the responsibility for the whole family.
7) His face took on a fierce expression
Fuel is taken on at the Frankfurt airport. We had to stop hal way to take on fuel

1) One boy had let off a firework in class. Many fireworks were let off on the saint patron’s day.

I'd use "set off" instead of "let off."

2) The boy was let off with a warning because he told the truth. The teacher let them off their homework. <~~Iffy, but ... OK, I guess! You’ve worked hard all week. I’ll let you off today.

Let on to:
3) I’ve told you a secret: promise you won’t let on. OK

It was very hot, and I took my sweater off. OK

4) John, take off your coat. Come through and meet the others.

Dad took the day off to come with me. OK

5)They took twenty percent off the list price. OK
The plane took off after refueling. OK
She became jealous when his career started taking off. OK

6) Don’t take on too much work – the extra cash isn’t worth it.

She has taken on the responsibility for the whole family. OK

7) His face took on a fierce expression.

Fuel is taken on at the Frankfurt airport. OK

We had to stop half way to take on fuel.

Better:

The teacher let them slide.

1) The sentences using the phrasal verb "let off" are possible, but they may have different meanings depending on the context. In formal English, you can replace "let off" with "released" or "allowed" for the first sentence, and with "excused" or "pardoned" for the second sentence.

1) One boy had released a firework in class. Many fireworks were released on the saint patron's day.
2) The boy was pardoned with a warning because he told the truth. The teacher excused them from their homework. You've worked hard all week. I'll pardon you today.

Regarding the phrasal verb "let on to," it does exist. It means to reveal or disclose something, especially a secret. In formal English, you can substitute it with "disclose" or "reveal."

3) I've told you a secret: promise you won't disclose.
It was very hot and I took my sweater off. (This sentence does not use the phrasal verb "let on to.")

4) John, take off your coat, come through and meet the others. (This sentence does not use the phrasal verb "let on to.")

Dad took the day off to come with me.
5) They deducted twenty percent from the list price.
The plane departed after refueling.
She became jealous when his career started to flourish.

6) Don't take too much work on – the extra cash isn't worth it. She has assumed the responsibility for the whole family.

7) His face assumed a fierce expression.
Fuel is refueled at the Frankfurt airport. We had to stop halfway to refuel.