Cutting the filament and feeding in more will not work 90% of the time.
The problem is that there is still some space between the extruder gear and the hotend. When the new filament passes the gear, the pressure will make it bend and slip off the old filament, and then your hotend will have filament stuck in it.
If this does happen, just remove the print head, set the printer to load new filament, and very carefully, push in something thin, like a 1.5mm allen key to push the old filament half way down into the extruder.
Remount the print head, set the printer to load filament again, and push in the new stuff.
if it isn't broken, you haven't modded it enough