If you receive a twenty day notice that lease will not be renewed and will have to vacate, and you need more time and willing to pay for another month can they deny you?

It depends. If you contacted your city, county or other entity to complain about leaky faucets, insects, broken step, etc, asking you to leave is then considered Retaliatory Action. Or if they gave you the note with only 19 days left in the month they can't evict you, the law is 20 days before the end of the month. If you haven't complained to a government entity then yes, they can ask you to leave with a 20 day notice. You also can give a 20 day notice, if you are out of your lease or on a month to month you can give them a 20 day notice. For the notice it must be posted either on the 10th of the month (in a 30 day month) or on the 11th of the month (in a 31 day month). It's tough to find an attorney to act on behalf of tenants, but if you offer to pay them for a letter in response to a Retaliatory Action they might do it, or go to a legal clinic. Landlords never listened to me, I had to have an attorney respond.

They can do what ever they want, all the rights they say renters have is bullcrap we have none