The 1.6 Turbo is probably a Gamma engine, not part of the theta II recall and extended warranty. Nonetheless, I know they have had issues with those too.
I suggest asking this on a Hyundai forum. Or, just tow it to a Hyundai dealer, assuming you have some decent maintenance records for the engine.
If Hyundai doesn’t fix it for you, I think a newer engine would probably not be worth it.