Actually, shooting with a compact camera rather than an SLR is part of the problem. Basically, shooting a game at night means that you don't have enough light for decent exposure times. As a consequence, your exposures are long and the moving objects are blurred. The field's lighting are rarely enough to allow you to capture moving subjects with ISO low enough so noise is not a problem. In this case, the options are to:
Add lighting (e.g., use a flash - but in your specific case there are 2 problems: your camera does not support an external hot-shoe flash, which may be solved by an optical slave, and more seriously, typical compact flashes just don't have enough power to light a soccer field effectively but it may be just fine for a race track if you can get close to the action).
Use a lens with large maximum aperture (small F-number). Your camera offers F2.8 which is not bad for this purpose, but in the SLR world you have lenses with bigger apertures.
Use a camera with more sensitive sensor. P&S cameras typically have far smaller sensors compared to SLR's. In the digital world, the bigger the sensor - the better its light recording ability. Pro sports shooters use full-frame, or at least APS-C bodies for the advantage in high-ISO low noise images they have over P&S cameras. You can dial up the ISO in your camera but the images will become very noisy/grainy vary fast. Also note that your camera is pretty old (2007 model?) which means that even for this form factor, you can expect better ISO performance from newer technology.
So, as you see, SLR camera will certainly help you to achieve better images in these tough conditions.
On top of all - remember to always use a tripod there and a remote release if possible. Even if the players and the ball end up somewhat blurred, at least the field and the background will remain sharp.