If she has paid for the week and the hours you are open then she has every right to send her child for time she legitimately owns...she has paid for it so it's her time. Many parents send their kids to child care so they can have time to run errands and do stuff at home, even if that something is take a shower alone or sit on the toilet alone...lol.
If she didn't bring her child but still had to pay for the week and you refused to take her child because she was home then I would say you have to give her her money back for that day. Either you have an agreement or you don't.
She paid for the time and it's her time. If you close early that's bad business. You need to fulfill your hours of operation. If you can't work one day you need to hire a substitute to come to your home while you are gone.
Since she's your neighbor I guess I'd let the kiddo come get the little one but if they weren't right next door then she'd have to come get kiddo herself. All of my parents received some portion of their child care through the state so they had to swipe their card every day or sign in late a day or two later. If they didn't swipe their card on time they got booted.
On another note, I've never, in all my massive years of child care ever ever seen anyone close before normal business hours.
I can't imagine you have many parents that can fit their schedule into your time frame.
A normal person goes to work at 8 and they don't work next door they have to drive and fight traffic. Most of my parents who had to be at work by 8am dropped the kids off by 7:30am. The kids ate breakfast with me so that worked out for everyone.
Then they all worked until 5pm. By the time they get out the door of the job and get on the road I had some that didn't get there until 5:30-5:45.
That's 10 hours minimum. You aren't even open 8 hours, how do you get any kids at all?
My point for asking is you seem to resent this mom for leaving her kiddo with you. She pays you for a week and then if she is off you don't want her kid to come.
Perhaps you could really examine why you do this business. Do you do it because you love being with kids all day and enjoy teaching them? Do you do it so you can stay home with your kids and this is the only way you can do that? Do you do it for another reason?
If you love this kiddo then you should be glad she's there every day. If it's just a job then perhaps you need to be open more hours so you can have enough kids to make it worth more?
As for the rest of it. Leaving their child with you on the weekend is over. You are allowing them to do this.
If they are not following the rules of your parent handbook then you need to do a form letter and send it to each of your parents, that way she can't say you picked on her. In the letter say something like you're changing some policies and want to make sure all the parents are aware of it.
Then say:
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Dear parent, I am changing a few of my parent policies and this is your official notification.
My hours of operation are 8am-4:30pm. If you are late there will now be a late fee. The first 5 minutes will be $10. Then each additional minute will be $1. So if you are 5 minutes late please be prepared to hand me, cash, $10.
Your child will NOT be allowed to come in my home until you have paid your late fee in cash. If you are late more than 1 time per month you face having your child dropped by me. I have activities with my children and we have a schedule to keep to.
Parents are to pay for the week BY drop off time each Monday. If you cannot pay on Monday and need to pay for the upcoming week on Friday that is fine. But your child will not be allowed to enter my facility until that week is paid for.
It is NOT my job to remind you to pay for child care. If you don't pay for your child care then you won't have child care and you won't get to go to work and then you'll be fired from your job, so pay for your child care first.
If you do not pay on time more than 1 time per every 6 months your child will be dropped.
When you pay by the week you have that time BUT when you tell me you are going to be gone, even for a day, I might plan take a drop in on that day. You will need to give me 24 hour notice that your child will be there.
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This will help this one parent to have that clear line of this business relationship. Otherwise you're going to have to fire them as clients and just be friends with them.