That tire needs to be junked right now
Head straight to the tire shop tomorrow morning, if they’re open
Any tire with cord exposed is unsafe and unacceptable. I agree with @Bing . . . you might be better off getting a set.
Another thing, have the mechanic balance the rim(s). Have him carefully watch while it’s spinning on the machine. If the rim is obviously bent, he’ll see it. Better to know as soon as possible.
And don’t let the shop talk you into nitrogen. If you’re not Nascar, it’s nothing more than a profit generator and bragging rights for the shop.
If the car has a tire pressure monitoring system, have them install a new kit (grommet, valve stem, aluminum nut, etc.). The old parts might not seal correctly, once they’ve been disturbed. The kit will only cost a few dollars more per rim.