There cannot be a set procedure for correction of an entry in the school certificate because every entry in the certificate is expected to be correct outright, particularly when the information is provided by you.
But if somehow some details happened to be wrong that can be corrected by the government department which issued the certificate based on your application along with the evidence to prove the matter.
Allowing correction of the certificate is normally discouraged by the department mainly because such corrections can be used by vested interest groups to add wrong data in the certificate even by producing fabricated evidence.
If the department is not ready to make correction in the record in your case, you must approach the High Court in a writ petition and get its direction to make necessary correction. It will be safe for the department to do so.
The high court can issue such a direction based on the principle that no certificate should carry any wrong data or details. The court is the final authority to deliver justice when the department is unwilling to do anything.