Preheat the oven to 350 degrees F (175 degrees C). Grease the bottom of a 9x13-inch baking pan.
Combine cake mix, water, eggs, and oil in a bowl; mix well. Pour into the prepared pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 23 to 28 minutes. Remove from oven and allow to cool until safe to handle, about 1 hour. Score cooled cake into quarters using a knife.
Line a baking sheet with parchment paper.
Remove 1/4 of the cake from the pan; cut off and discard any hard outer edges. Crumble cake into a bowl and add 1 tablespoon icing; use the back of a spoon to mash until mixture binds together. Use a cake-pop baller or small ice cream scoop to shape mixture into balls. Roll balls between cupped hands until firm and smooth. Place on the prepared baking sheet. Repeat with remaining cake quarters and icing.
Cover cake balls with plastic wrap and foil and refrigerate for at least 4 hours. Cover a styrofoam block with foil to use as a stand for the cake pops; use a knitting needle to punch approximately 40 holes in the block.
Remove cake balls from the refrigerator. Use a cake-pop baller or thin knitting needle to make a guide hole halfway through each ball. Return cake balls to the refrigerator while preparing icing.
Place icing in a microwave-safe bowl and heat in the microwave until softened, 30 seconds to 1 minute.
Remove cake balls from refrigerator. Push the end of a lollipop stick into a ball; dip ball straight down into icing until submerged and melted icing touches the stick. Tap softly to remove excess icing, and immediately cover with sprinkles. Push cake pop stick into the styrofoam block. Repeat with remaining cake balls. Allow to dry, about 1 hour.