You need to set the schedule firmly, as well as the budget. It is your job to set the limits, not hers.
Who does this? A college kid who has been told to entertain a 5 year old. In college, I took care of a 6 year old two nights a week while her mom was in graduate school. I got her off the bus and was with her until well after bed time and the expectations of me were very clear from day 1!
Safe... happy... clean... fed. I had a key, so the door was to be locked. She had toys and games (not to mention homework), so we had things to do. Her toothbrush, tooth paste and jammies were ready to go before mom left, so she was clean. I enjoy cooking, so there were always ingredients and a recipe card on the counter, so she was fed. No need to take her anywhere.
Just a thought...
If you don't want them going out to eat, make sure that there is enough "fun food" in the house for the day, including popcorn and other OK snacks. Leave a list of OK activities and a $20 bill on the table as the "budget" for the day. Be clear with her that this is their "fun money" for the day, nothing more. I would also leave a schedule out for them.
8:30 Breakfast
10:30-11:00 Table top activities (puzzles, coloring, play doh, letters, numbers, crafts, etc)
Noon Lunch
1:30 Outside play
2:00-2:30 rest time (if it's OK, leave out a movie for your LO to watch
3:00 Snack
If you look at the day this way, there's very little chance that she'll need to venture out. At your child's age, a preschool or daycare provider would structure the day this way and so should you.
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