How is it possible for the First Order to be recharging the Starkiller? If it fired when the sun was depleted, do they have a spare sun? It seems to me that it was designed as a grand single-use weapon, or alternatively it would need to be moved after each use.
Think of it like this: when you charge a battery, do you use all of its energy at once, or do you use chunks of it at a time? That’s the same way Starkiller Base works in this edit. They used up a lot of the energy to destroy the Hosnian system. It then took some time to recalibrate the weapon to fire on the Resistance since it just ate up a lot of the energy. But my point is that there was some energy left after firing.