Public keys can also be encrypted, but it will be useless if the private key of your encrypted information is not correct. By the way, both public and private keys can be encrypted and decrypted.

I don’t know what the exponent you are referring to refers to. To calculate the public key, you must know e. Now that the module is known, we only need to know e. e is a randomly generated number between 1 and φ(n) and relatively prime to φ(n). Since you don't know φ(n) now, you can try using e to participate in the operation to see if it is correct. If it is correct, then the exponent you are talking about should be e. You can give it a try.