Need advice for 'stray' i'm feeding

We often feed cats in the neighbourhood that look like there isn't anyone taking care of them. Recently a small (most likely a kitten)cat began coming around. Her tail is missing but she had a collar. She refused to let us come near her at first and we just left food on the patio for her.
She began coming up on the patio and crying for food about once a day afterwards. About a week later and she was allowing us to come out and sit on the patio while she eats.
I just discovered she belongs to a family two houses down from us. That's not far since we live in Duplex's on this street.
They have two 4 year old boys and I'm pretty sure there is another cat in the house. I don't know them at all but the one of the little ones asked me if I'd seen his cat. I told him I had and told him to leave food on his patio to get her to come home. I don't know if they did or not.
Why won't she go home?
I stopped feeding her for about 3 days. She disapeared and then reappeared very much disheveled and dirty and skinny.
I don't know if I should keep feeding her since it doesn't seem to make her go home. But I'm having trouble to try and stop feeding her as I don't want her to be hungry. What should I do?

She obviously isn't being cared for by that family so I personally would get her and either keep her if I could or find a no-kill shelter to take her to in hopes she would find a home with people who care. I did that to a cat that was coming to our house. He now has a wonderful home with caring people.

You could / should go and talk with the adults in this house. Friendly of course.

You cant go through a 4-year...

It may even happen the cat isnt at all theirs. It may be so the child got fond of her, so he called her for his cat.

During the time it clears up, you can get hold on her and help her. Releasing first when everything is clear and the neighbour giving clear promise of taking care.

IF it happens she isnt really theirs:
Does she still have her collar on? another scenario may be to attach a message to this collar, with your phone number. and wait for answer. But here prob better to take her in, and set up messages "Found" around.

Good luck!

I would go and speak to the people and just tell them you will take the kitten and find it another home or keep for yourself. But try to talk with them first. If they don't care or if this kitten keeps coming over to stay, then by all means save this kitten. Take the kitten inside. And try to find another home for the little sweetie maybe even a rescue/foster situation if you can't keep the kitten yourself. Thanks for caring and hope you can get this resolved quickly for the kittens sake and safety.

I would do exactly what you are, feed it and try to give it some positive attention. The poor thing. So glad you are there to help!