AF-ON on D600 ... hit and miss. Me or the camera?

If focus ends up so far off, it's usually user error. Make sure your focus point is on top of the subject when you press AF-ON. As the subject moves, move the camera to keep the subject under the focus point. Dynamic area gives you a little leeway if the subject is really hard to track, but if your selected focus point is off the subject for too long, it'll refocus on whatever the focus point sees. And that could very well be the background.

If you're having a really hard time keeping the focus point on the subject, you can try switching to 21 point dynamic. It'll give you a little more room for error, especially if you're framing tightly (reducing the chance that the camera will focus on another player also covered by the 21 points).