Diagnosis
Yeah, this is a pain. The dome is plastic (not glass) and is coated to repel water. Using the wrong cleaner or cloth will smudge that coating and make the video permanently hazy.
How to Fix It
Power the camera down first to be safe. Use a clean, soft microfiber cloth—the kind for eyeglasses or camera lenses is perfect. Dampen it slightly with distilled water or a 50/50 mix of distilled water and white vinegar. Wipe the dome gently in a circular motion. Immediately dry it with a second clean, dry microfiber cloth. Avoid paper towels or tissues, they scratch.
If That Didn't Work
For stubborn grime like sap or bird droppings, use a tiny drop of mild dish soap (like Dawn) in a bowl of distilled water. Dampen the microfiber cloth in the solution, wring it out well so it's barely damp, wipe the spot, then immediately rinse with a cloth dampened with only distilled water, and dry thoroughly.
Heads Up
Never use window cleaners, ammonia, alcohol, acetone, or abrasive cleaners. They will permanently damage the anti-glare and water-repellent coating on the plastic dome. Once that coating is gone, you'll get worse glare and water will bead in a way that distorts the image. Also, don't spray anything directly onto the camera; spray onto the cloth.