can the little spare tires,of todays cars. be repaired,or just buy a new one.

It depends on where the hole is and how large it is. If it’s in the side wall, then it can’t be fixed. If it’s in the tread, it can be fixed if it isn’t too large. Why do you need to fix it? They are only supposed to be used to get you to the shop, not driven on for any length of time.

These are considered throw away tires, so get a new one.

Dollars To Donuts
Your local auto recyling yard should have one (like-new tire/rim) for you for a few dollars. Make sure it’s right for your car before you need to use it.