Cut out the damage and install new ends. There is no safe temporary fix.

The only proper permanent repair for a cut or damaged extension cord is to cut out the damaged section and install a new male or female plug (or both, depending on where the damage is). Cord repair kits exist for intermediate splices, but most old cords have internal damage that makes them unsafe long-term. New ends are cheap (a few bucks each) and dead simple to install with a screwdriver. Do that.

As for temporary: don’t. Electrical tape will eventually slip, heat up, and start a fire. Wire nuts with tape? Not rated for repeated flexing. If