manticle
Standing up for the Aussie Bottler
Proteins forming cold break will drop out. You can leave these behind (I don't but I ferment straight in the cube and plate chillers don't - it won't do harm to the brew but may cause chill haze. There are other ways to prevent chill haze). No need to shake it up although a certain amount of the break material can act as a yeast nutrient.
To answer the second question - yes you can reprime but I wouldn't just add a carb drop. Calculate how much you were originally short by and divide that by the number of bottles you have. Multiply that rate by the number you have left, dissolve in solution (boiled water) and syringe the exact dose into each bottle (make the water volume easily calculable like 10 mL per bottle for example).
To answer the second question - yes you can reprime but I wouldn't just add a carb drop. Calculate how much you were originally short by and divide that by the number of bottles you have. Multiply that rate by the number you have left, dissolve in solution (boiled water) and syringe the exact dose into each bottle (make the water volume easily calculable like 10 mL per bottle for example).