Can you repair a cut extension cord?

Yes, but it’s probably not worth your time. The safest way is to cut out the damaged section and install a new male or female plug end, which costs a few bucks at a hardware store. For a cut in the middle, use a waterproof junction box or inline plug kit — never just twist wires and tape them. But honestly, if it’s a cheap cord, just throw it away and buy a new one for $15-$20.