Dear Chethan, 

Mop Up round will be conducted after the second round of counselling in Karnataka, for giving admission to seats which remain vacant after the second round. In this round, while all the NEET qualified candidates who register for the counselling will be eligible to participate, the seats will be allotted in the order of merit as per the categories. So, there is no guarantee of securing a seat, since seats will be allotted as per merit. 

Coming to the other part of your question, the mop-up counselling round will take place for both government and private seats. 

Hope it clears your query. 

All the best!