No, A government employee is not allowed to run a private business, neither is he allowed to work anywhere else as a part-time or full-time employee. This is against government rule and hence person who is found can be charged for breaking the law. Government employees are bound by job contracts with the govt and are not supposed to work part time.
It is written in the conduct rules of the govt that any govt employee is not allowed to hold any office of profit/another job while still in govt service. In order to do another job you have to resign from the govt job.
You could apply for another job. However, if you apply without the permission of your present employer, it could cause troubles at a later stage. You will have to resign from the present job to join a new job. Your past service and related benefits will not be carried forward to your new job. In government jobs every department and every scale of employment has its own set of terms and conditions. However normal freelance work is normally not barred. However the specefics can only be ensured after looking at your appointment conditions.
As per government services Act, government employees can’t indulge in private businesses like teaching in tuition centres. Recently, the department again issued a circular saying no teaching faculty of School Education shall take up any activity in private institutions or coaching centres.