Joint tenancy - one tenant left but its in statutory period

Tenancy Types and Management

I have recently taken over management of a number of properties on behalf of the landlord. The previous agent left the portfolio in a bit of a mess.

One property has a joint tenancy, a 12 month AST, commencing 31/05/21. So, the tenancy is now in statutory periodic contract. However, one of the tenants moved out in November 2021. The other tenant is still in the property and has taken on the full payment.

However, what I am concerned about is that both names remain on the tenancy, so essentially they are both still legally responsible even though the tenancy is now statutory periodic.

The remaining tenant doesn't want to take out a new AST. They want to remain in a rolling month to month contract. What can I do to rectify this? Is there such a thing as a addendum to the original contract?