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.