The plastic is probably just a protective cover over the wires in the electrode, the electrode is the glass end of the tube and is melted onto the tube. If it's just broken, a new electrode could be fit on the tube, but that's something a full on neon shop would have to do.
You have to trim the broken glass off perfectly straight and then "weld" the new electrode cap on. Then pull as close to absolute vacuum as possible on the tube while heating it up with a high voltage and current to cook the impurities out. Then, while still under vacuum, you let just a touch of the gas you need for the color desired. Melt the little vacuum tube connection on the electrode shut and you're done!
Way more info than you needed, just thought some might be interested in the short version