Place berries and 3 tbsp. powdered sugar in a blender or food processor and roughly puree. Taste and add more sugar if desired. You should have about 1 cup berry puree, and you can make it as smooth or as chunky as you like.
Transfer puree to a bowl or measuring cup with a pour spout and set aside. Wash out blender or food processor and then place cream cheese, half-and-half, vanilla, lemon juice and ½ cup powdered sugar inside. Puree until smooth.
Fill Popsicle molds, alternating berry mixture and cream cheese mixture. If desired, stir gently with a Popsicle stick for a swirled effect, or leave layered. Freeze until firm. Add a wooden Popsicle stick to each mold and then freeze until completely solid.
To remove from molds, carefully run the outside of the mold under warm water until popsicles are loosened. Store in an airtight container or a zip-top bag in the freezer.