State quota seats & private colleges counselling will be conducted by the particular state counselling authorities. To apply for the other state medical colleges you will be eligible for the management quota seats in private colleges. If you wish to take admission in Karnataka & MP medical management quota seats then you can register for the second round there is one application form for Government & private colleges also. You can check the further information with the given below links.

https://medicine.careers360.com/articles/mp-mbbs-counselling

https://medicine.careers360.com/articles/karnataka-mbbs-counselling

Good Luck!