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Solution #1: Go into Settings->Game Center and log out of your Game Center account. Reboot the device, and re-run the game and see if that fixed it.
Solution #2: Do #1 above, but also delete the game prior to rebooting. Reinstall and try again.
Solution #3: There is a bug in Game Center where without warning it will simply stop working on specific networks, so try running the game on a different network. If you're on WiFi try it on cellular and see if it works, or vice-versa. Also, try it on a friend's WiFi network or maybe the network at work or at school.
Solution #4: Go into Settings->General->Reset to reset all of the settings on your device. This can also sometimes fix Game Center problems.
Solution #5: Wipe your device clean and start over. This is drastic, but sometimes it's the only way to fix Game Center. You'll need to restore the OS on the device, but not restore the device from any recent backup. If you restore from a backup you're just restoring the problems, so you have to start clean.
If none of the above works then you're one of the lucky people who Game Center simply doesn't work anymore on iOS 7. Please contact Apple and complain to them, as we've been telling them about these bugs for ages, but they never fix them.
Switching to Gyroscope-only mode should get rid of the "pops" in the controls, but without the compass the gyroscope will tend to drift out of calibration, and you'll have to press the Calibrate/Neutral button in the game pretty often.