How can I eliminate motion blur with a Canon Powershot A560?

Use a higher shutter speed of around 1/250 or higher. This will require you to use either a higher ISO speed, or a faster aperture. High ISO speeds will give you grainy images, though. To solve this, use the fast shutter speed with a flash. This will allow you to keep the ISO low.

auto use "sports" mode

manual, iso of 400 or so, shutter of 125th or so

flash will also freeze most motion as long as you are close enough - depends on flash power etc

a

Read & study the Owner's Manual. That is the first step in getting good results with your camera. Cameras do not possess "magical" powers that enable them to take good pictures. Good pictures are a result of knowing how to use the camera. Its the photographer, not the camera.

You are likely experiencing that bane of digicams known as "shutter lag" - the fraction of a second between when you press the shutter button and the shutter actually responds. You can try pressing the shutter button 1/2 way down so the camera can pre-focus and then finish releasing the shutter.

faster shutter, higher iso and bigger aperture to make sure enough light