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Whisk sugar to taste, cornstarch and salt in a large saucepan. Add milk and whisk until combined. Bring to a boil over medium heat, whisking occasionally. Boil, whisking constantly, for 1 minute. Remove from the heat and stir in vanilla.
Put about half the pudding in a food processor or blender and add bananas. Process until smooth. (Use caution when pureeing hot liquids.) Stir the mixture back into the remaining pudding.
Divide the mixture among freezer-pop molds. Insert the sticks and freeze until completely firm, about 6 hours. Dip the molds briefly in hot water before unmolding.