Hi..
Welcome to HEALTHCARE MAGIC..
I have gone through your query and can understand your concerns..
As per your complain it seems that white chunks coming from throat and having discomfort associated with tonsils is indicating towards a condition known as Tonsil stones or Tonsilloliths..
They are formed when there is calcification of food debris and dirt that gets accumulated over the tonsillar crypts and it leads to formation of small calcified chunks that can come to mouth by dislodging from the tonsils on coughing..
It is a potent cause for recurring 
throat infection and 
tonsillitis..
I would suggest you to consult an Otolaryngologist and get evaluated and a thorough clinical evaluation is generally sufficient for diagnosis..
In case of tonsil stones you can start doing 
hydrogen peroxide gargles by making a solution in 1:1 ratio with water..
Do warm saline gargles.
Always gargle with atleast normal water after eating anything..
Limit intake of dairy and other high calcium foods..
Hope this information helps..
Thanks and regards.
Dr.Honey Nandwani Arora..