Singer motors I have had with this configuration use wire nuts to connect the wires in the housing under the motor, which makes replacement of wiring a whole lot easier. If you have a button type foot controller then it likely has two screw on wire terminals inside of it for that end of the wire, again, a whole lot easier to replace. When I replace wire to anything but the lamp I use wire of a heavier gauge than lamp cord because of the additional power draw of the motor. And I am guessing that the latch on that replacement plug isn't closed all the way because the wire coming in is the original wiring (heavier than lamp cord) and that plug is made to clamp onto lamp cord. You want an appropriate replacement plug for the heavier gauge wire used there.

I took off the sketchy patch and it's not a break, just a cut in the insulation. I think I'll just clean it up for now.

Thank you, it does appear that the cord is heavier than a lamp cord, I can take it off and get one with screw on terminals at the hardware store. It looks like the wire is the original coming out of the motor. Maybe that's the only changes they made?

Hard to say without seeing it all in person, but my guess is the wire into the original plug was ruined by someone pulling it out by the wire instead of the plug. And remember do not leave your machine plugged in when not in use.

Picked up a proper size plug today but I'm waiting on the shrink wrap I ordered before replacing it. I want to cover the cut in the insulation with shrink wrap before putting the plug on.