1. Place the lemonade, rum, lychee and honeydew in a food processor or blender and process or blend until puréed, about 1-2 minutes.
2. Pour mixture through a medium strainer.
3. Pour the strained mixture into popsicle molds—add raspberries now if you are garnishing the popsicles—and freeze for about 2 hours or until mixture starts to solidify enough to hold a popsicle stick upright.
4. Insert the popsicle sticks and finish freezing popsicles overnight.