What hours are reasonable to ask caregiver, when state only approved 20 hours per month?

Did the state suggest what agency to use? You could talk the possibilities over with them.

One hour a day for someone who lives close by or who has other clients in the general neighborhood might be feasible. Most people would find the transportation issue a problem. Driving any distance or taking a bus for one hour's wage may not be practical. If there were a couple other clients in the same apartment building or very nearby, that would make a difference.

Talk to the agency.

Yes, talk to your local agencies, it is my understanding that some of them have minimum hours limits. Where I live in Canada with government funded home care cut backs have caused many to have to get by with only 1/2 hour - I've even heard of 15 minutes , a ridiculous extreme that satisfies neither care recipient nor employee.