It depends. Whether you actively renew or auto-renew, the process will work the same way.
If your grace period spans the calendar year, your insurance company will send you a bill for your January premium and will also bill you for your past-due premium debt. If you only pay the January premium, the insurance company can apply that payment to the amount of premium debt that you still owe. Unless you catch up on your premium debt and pay the January premium by the end of the grace period, your insurer can cancel your coverage for next year and retroactively terminate your current coverage to the last day of the first month of your grace period. This will be the case whether or not your insurance company has adopted the past-due premium collection practice.
If your premium debt started earlier this year, so the grace period already ended, then it will depend on whether your insurance company has adopted the new premium-debt repayment practice. If it has, it can apply your January payment to your past-due premium debt and require that you pay any remaining past-due premium and also pay the January premium before it will let your new coverage take effect. The insurance company must have provided you notice that it is adopting the premium-debt repayment policy. The notice must explain the deadlines for paying premiums and repaying premium debt and the consequence for not meeting these deadlines. Contact your insurance company if you have questions about this notice or your bill. You can also contact the Marketplace or your state insurance regulator if you have questions or concerns about this new policy.
If your insurance company has not adopted the new debt repayment practice, then it may accept your January payment to start your new coverage.