Replace a damaged extension cord plug, don't patch it.
Don’t try to repair a cracked or broken plug by taping or splicing. Replace the whole end.
Extension cord plugs and receptacles fail for a reason—strain, corrosion, or physical damage. Taping a cracked shell or reusing a loose prong invites arcing and fire. A new plug end costs a few bucks at the hardware store and installs in five minutes with a screwdriver. Same goes for the receptacle end: cut off the damaged part, strip the wires, and screw them into a new connector.
The only safe “repair” is a clean replacement. If the cord itself is frayed or nicked, cut back to good insulation and replace the plug. If the damage is near the middle of the cord, don’t splice it—buy a new cord. Spliced extension cords are a headache and a hazard.
Safety here is not that complicated. Just don’t half-ass it.