An insult, akin to "stick that in your juice box and suck it". From the video gameAnimal Crossing, where furniture turns into leaves that can be put in your pocket and retrieved later.
Instead of covering your mouth with your hand and coughing a person can cough into their folded elbow area. This creates a cough pocket. It also keeps you from spreading germs on your hand.
John had a terrible cough. He kept coughing into his hand. His uncle said, "Cough pocket John! Cough pocket!"