Coughing up white tissue with tinge of blood. What is it?

The usual mucous . activated by ingested virus or bacteria; normal mucous produced by lungs to trap inhaled microscopic material is white; becomes either yellow or green if enough WBCs are present ingesting virus/bacteria & making internal enzymes (producing the color) used to inactivate & digest viral particles or bacterial cells. Repeated intense cough can break blood vessels resulting in tinges of blood.
White=thick phlegm. I would advise you to see your doctor regarding your cough with white tissue and tinges of blood. The white tissue is likely some form of thick phlegm. The tinge of blood is likely from excessive coughing to bring the phlegm out. A physician needs to have a good look at you in order to determine the level of treatment that you need to overcome this aliment.