Diagnosis
You're right to be careful. These screens have a matte anti-glare coating that can be damaged by harsh chemicals or rough cloths.
How to Fix It
Here's the safe method. First, turn the TV OFF so you can see the smudges. Use a dry, ultra-soft microfiber cloth (like the kind for cleaning glasses) and gently wipe in a straight line across the screen. No circles. If that doesn't work, dampen a second microfiber cloth with *distilled water* only. Wring it out so it's barely damp, not wet. Wipe the screen gently, then immediately follow with the dry cloth to buff off any moisture. That's it.
If That Didn't Work
For stubborn, sticky spots, use a 50/50 mix of distilled white vinegar and distilled water. Dampen the cloth with the mix, wring it out thoroughly, wipe the spot, and then immediately dry it. The mild acidity can break down sticky residue without harming the coating.
Heads Up
Never spray liquid directly onto the screen—it can run down into the bezel and damage electronics. Never use window cleaner, isopropyl alcohol, paper towels, or any abrasive cloth. They will permanently scratch or haze the anti-glare layer.