Diagnosis
You're right to be cautious; using the wrong cleaner can permanently damage the anti-reflective coating on the screen.
How to Fix It
Use only a dry, soft, lint-free microfiber cloth. Gently wipe the screen in a circular motion. For stubborn spots, slightly dampen a second microfiber cloth with *distilled water only*, wring it out completely so it's barely damp, wipe the spot, then immediately dry the area with the dry cloth.
If That Didn't Work
For persistent, non-greasy smudges, you can use a 50/50 mix of distilled white vinegar and distilled water applied sparingly to the cloth, never directly to the screen. Dry immediately. Test in a small corner first.
Heads Up
Never use window cleaner, ammonia, alcohol, or abrasive cloths (paper towels, tissues). Never spray liquid directly onto the screen, as it can seep into the edges and damage the panel.