Hi.. Welcome to HEALTHCARE MAGIC.. I have gone through your query and can understand your concerns.. As per your complain the white hard substance with tinge of blood over it that you have coughed out seems to be Tonsil stone or Tonsillolith and is formed due to calcification of dust and food debris that gets accumulated over the tonsillar crypts and it is a potent cause of recurring sore throat and bad breath.. I would suggest you to consult an Otolaryngologist and get evaluated and in case if more stones are present attached to the tonsils he can remove it.. To prevent further stone formation you can gargle with warm saline water twice a day, gargle after eating anything and limit intake of dairy Products.. For sore throat take a short course of antibiotics like Levofloxacin and anti-inflammatory painkiller like Aleve and do warm saline gargles.. Sucking throat lozenges can also relieve.. Hope this information helps.. Thanks and regards. Dr. Honey Nandwani Arora.