Preheat oven to 400 degrees F (200 degrees C). Grease and flour a 9-inch cake pan.
Combine 1 cup sugar, egg, and 1 tablespoon butter in a large bowl; beat with an electric mixer until creamy. Beat in apricot jam, vinegar, ginger, and baking soda. Add 1 cup milk and flour; beat until batter is smooth.
Pour batter into the prepared pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 35 minutes.
Combine 1 cup milk, 1/2 cup butter, 1/2 cup sugar, and water in a saucepan over medium heat 5 minutes before pudding is done baking. Bring to a boil and stir until smooth. Pour hot sauce over pudding and let it sink into the pudding.