What could cause ice pops (popsicles) to go soft in the freezer?
Your freezer is set at too high a temperature, probably just below the freezing point. Pure water freezes at 32°F/0°C so your meats and vegetables are freezing just fine, however adding sugar and/or salt to water reduces the freezing point. A 30% sugar solution freezes at around 28.5°F, salt water is about the same. The fact they are mushy means your freezer is almost, but not quite cold enough to freeze your popsicles.
Frozen desserts aside your freezer is far above the recommended temperature, the FDA and virtually every other guideline I see says that freezers should be set to 0°F, which is -18°C, not because of the freezing point but because that's the highest ideal temperature for food to last long-term.