Platelets, also called thrombocytes (thromb- + -cyte, blood clot cell), are a component of blood whose function (along with the coagulation factors) is to stop bleeding by clumping and clotting blood vessel injuries.
Milk of the poppy is essentially the Westeros version of morphine. Maesters make it from crushed poppy flowers, and morphine and other opiate-based drugs are made from poppy plants. Indeed, the books' descriptions strongly suggest that the poppy is intended to be the real-world opium poppy (Papaver somniferum).
However, it is known that morphine and codeine can sometimes be detected in the urine up to 48 hours after ingestion of poppy seed from some pastries such as bagels, muffins, and cakes (Reference 1 is a very interesting FREE article on this topic).