How do you repair a severed power cord to an outdoor heat pump?

First you need to ask yourself how it got severed and if a cord is even a good idea. Maybe a different installation would be safer and more durable, avoiding further damage to the cord. You may replace the cord or splice an approved type of cord with a kit that includes an approved splicing device, for the proper size wire, and an insulation device that will be as good as the original insulation. Or you may strip, join and twist, then solder each of the wires and add sufficient insulation to each one and then to the splice itself. Poor splices are responsible for many fires. Under the National Electrical Code, you cannot repair (splice) a power cord unless it is a "hard service" or "junior hard service" type (i.e., S, SO*, SE*, SJ*, ST*). A spliced cord cannot be re-used in a different installation. It may be a better idea to replace the entire cord anyway. You may not repair the following types:portable power cables (types G*, P*, W), heater cord (H*), vacuum cord (SV*), dryer and range cords (SRD*), or lamp cords (C, SP*). These unapproved types must be replaced from end to end if they are damaged. :The * means any other letters or numbers that appear in the type label on your cord after the prefix (e.g., SP* includes SPT-2W, SPT, SP-1, SPE).