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Hi there. The blood thinner obviously will make it more likely that you will bleed from any small break in the skin or mucous membranes of the body. Blood that streaks or otherwise comes out with mucous could originate from a broken blood vessel in the nose (that is draining down into the throat), or one in the back of the mouth or the airway. The forcefulness of a cough can be strong enough to break a small blood vessel as well. Blood or mucous from above, or acid reflux from below (heartburn) can also cause an irritation in the throat that feels like a lump or foreign body that one cannot clear with coughing. While it is unlikely that the medication itself is causing this sensation, it is worth mentioning to your family doctor, as some medications can trigger an irritating cough...one that typically clears within a couple weeks of stopping the medication.
If this problem worsens in any way, or your doctor is concerned, he can refer you to an ENT specialist who can look down your throat with a special, thin flexible camera to see what is going on.