The rules of Mop up round are simple.  Eligible Candidates for Mop Up round:

1. Candidates who registered for Round 1 and opted for Free exit

2. Candidates who registered in Round 2, were allotted a seat and exited with forfeiture of fees

3. Candidates who did not register for Round 1 or Round 2,

4. Candidates who registered either for Round 1 or Round 2 but did not get a seat

Candidates who have already joined in Round 2 of AIQ/Deemed/Central Universities/AIIMS/JIPMER are not eligible for Mop Up Round. https://mcc.nic.in/UGCounselling/Documents/UGSchemeFinal.pdf

If you registered for Round 2 and then did not get a seat, you needn't forfeit your security deposit because you were not assigned any college to join. And so you are eligible for appearing in Mop up round. And you do not need a fresh Registration. You can use the same registration credentials to appear for Mop Up round Counselling.