Preheat the oven to 350 degrees F (175 degrees C). Grease a fluted tube pan (such as Bundt).

Peel mango with a vegetable peeler or paring knife. Cut flesh into small pieces and squeeze pit for juice. Collect all juice from chopping and squeezing pit and reserve for glaze. Smash mango in a bowl using a potato masher or fork. Measure 1 cup smashed mango, reserving the rest for the glaze.

Beat 1 cup smashed mango, cake mix, eggs, water, vegetable oil, and lime zest together in a bowl using an electric mixer for 2 minutes; pour into the prepared pan.

Bake in the preheated oven until a toothpick inserted in the center comes out clean, 40 to 50 minutes. Cool cake in pan for 5 minutes. Invert cake onto a wire rack to cool completely.

For the glaze: Combine melted butter, lime juice, rum, and about 1 tablespoon reserved juice from mango in a bowl. Stir confectioners’ sugar into butter mixture until desired consistency is reached; fold smashed mango into glaze. Spoon glaze over cooled cake.