Stir the cornmeal, sugar, salt, and water together in a mixing bowl until a dough has formed. Set aside to cool for 10 minutes. Once cool enough to handle, place a spoonful of dough into the palm of your hand; make a depression in the center. Fold a half slice of American cheese into a wad and place into the hole. Spoon some more dough overtop; roll the corn stick into a sausage shape, making sure that the cheese is not exposed. Repeat with the remaining dough and cheese.
Heat oil in a large skillet to 375 degrees F (190 degrees C).
Cook the corn sticks in the hot oil until they turn golden brown and float to the top of the oil, about 4 minutes per side.
Drain on a paper towel-lined plate before serving.
