I would write a registered letter, return receipt, and inform the mother that your dog is a guard dog and trained to attack anyone that enters your yard. That you have signs posted stating this. That her child has entered your backyard and each time they do they are risking their life. If the child will knock on your front door you would be glad to go back and look, at your convenience, for said toy item and then toss it back over the fence. But main point being that her child could be killed for entering your yard.
On one hand, you've posted signs. YOU know your dog could injure other people. YOU are liable for what that dog does. You might speak to an attorney about this situation because, from what I've seen on the news, IF you know your animal is a threat to others and others are hurt by your animal you can be held liable if your dog hurts someone, even in your own backyard.
I would not live where you live. It sounds like a slum lord area. People who live like you said above aren't good, steady, parents. They sound like really bad parents who don't care.
I would give notice, if you're renting, and move as soon as I could.
If you purchased this house you could put up several signs that say no trespassing and that violators will be prosecuted or something.
Then I might go as far as putting barbed wire/razor wire along the top of my fence to keep the neighborhood kids out and then I'd do something like tall shrubs around the perimeter of my front yard and put a gate across the driveway. I'd barricade my property.