One little tweak that will instantly make your child a better singer

Sing higher.

Not louder - higher. Find a high, light delicate sound:

You don't have to sound exactly like a boy soprano, but getting in the ballpark will help.

See, the natural range in which American adults speak is rather low. Inexperienced singers will pitch their voices in the range of speaking, since that's what they're used to.

Can you see the problem? Children's voices are higher than those of adults. However, the kids will go with you into the basement if that's where you start the song off, and they'll sound terrible because they won't be able to hit the notes.

If you move the song up in key, all of a sudden the kids' voices will sound sweet and their pitch will be far more accurate. It's amazing!

You might feel as though you're singing way too high. However, if you're singing a simple chidren's song like "Are You Sleeping?" the range is so small that you're unlikely to run into trouble.

Try it, and let me know how it goes!