OK
1. Forget the white background unless you can use extra lights - but then you are giving yourself more to think about and you don't need that.
2. Get the subject as far away from the background as possible. If you use the flash with a shoot through umberella and shoot say 3/4 to head and shoulders type shots then the background will be pretty much black.
Set the camera and flash to MANUAL. Put the flash to say 1/4 power, ISO to 100 and shutter speed to 1/125
That should block out all the ambient light. Now adjust the aperture until you get the exposure right on the camera. Try f8 to start. If too bright go to a smaller aperture like f11 or smaller or perhaps somewhere in between.
The flash power is controlled by the aperture. Just adjust the aperture until the flash looks good on your camera (checking the histogram will help too to make sure)
What you should get is a black background with your subject exposed correctly. It's important that the light you use does not light the background - that's why the subjects need to be as far away from it as you can get them (doesn't need to be too far but far enough away that the light doesn't register). If you see light on the background (in your camera), just move the light a little closer to your subject and adjust the aperture to suit.
This was window light in a hotel room and the bride was very close to the window. The closer the light is to the subject the quicker the light it will fall off which renders the background black. There was a bed, people in the room and the wall was only about 10-12 feet away...... Not a perfect shot but used to show that you can eliminate a background easily.
I could complicate things by saying read up on the inverse square law - But that just screws with the mind
A white background or using a sheet of some kind will likely just look messy and distracting.
Also I appreciate it's kids you are shooting and they can be tough. Doorways work well - Nice soft directional light that falls off quickly and provides a near black background if the hall behind is long. You may not even need the flash!