Hello, I am an attorney and am happy to assist you today.
If the lease is expired, you don't have to allow the tenant to stay any longer. If consistent payments are an issue, it's time to ask them to leave.
Once the lease is over, the law does afford the tenant the right to stay, but only on a month to month basis, and if they pay their rent. So you need to give the tenant a notice asking them to move out by the end of August. You have to give a full 30 days notice from the first of the month, so that's why it's not until the end of August.
Then, once you provide that notice, you also have a claim against the tenant if they don't pay their rent. In that case, you can also evict them even before the end of August. But, it's important to go ahead and get that notice sent out to cover yourself, so you know for sure that you will be getting rid of this difficult tenant.
Please let me know if you have any questions about this.