How to patch leaky guttering.....

You could try Gaffer Tape for a Temp fix. I would put it both sides.

you can buy a gutter patch from most big box hardware stores or you can cut a tin can a little bigger than the hole. Flatten it over the hole and epoxy/seal it.

You can use exterior caulking (both sides) for a budget friendly quickly repair. Be careful. Best of luck.

There are cans of spray stop leak you can apply. Home Depot or Lowes should have it. You apply to the inside of the gutter. I think it comes in black or white, and made for this kind of thing.

There are also patches that you can stick on the leaks.

I used a rubber caulk the True Value salesman recommended and a caulk gun for a couple pin holes in my gutter.