AYC can't help if you lift off on a corner. As soon as there's no power, it's powerless.
Only the RS came without AYC, and they're pretty rare, you're FQ will be based on a RSII i think, which has AYC. You can have a look in the little compartment on left in the boot (just behind the wheel), that's where the AYC reservior is. If is empty someone's swapped your diff.