HS student got expelled immediately from private school for possessing marijuana first-offender. Can he return to school and also erase the record?

Private schools do not have to follow State regulations.  If the school has a policy of zero tolerance, and the policy is to expel any student that is found in possession of drugs, then the school can follow that policy.  In the public shcool, it might e a first offense, and it might not result in expulsion, but a private school gets to make their own rules.