Diagnosis
You're right to be careful. Those little dark circles around the main lens are the infrared LEDs for night vision, and smudges on them or the lens will ruin your picture quality.
How to Fix It
Power off the camera by unplugging it. Get a soft, clean microfiber cloth (the kind for glasses or screens). Gently wipe the entire front face of the camera in a circular motion, applying almost no pressure. You can lightly breathe on the lens to create a bit of moisture to help lift dust. Do not use any liquids, window cleaner, or compressed air directly on the lens. The compressed air can force dust into the seams and the chemicals can damage the plastic lens coating.
If That Didn't Work
For stubborn spots, you can slightly dampen a corner of the microfiber cloth with distilled water, then wipe the lens gently, and immediately dry it with the dry part of the cloth. Never spray liquid onto the camera itself.
Heads Up
Avoid touching the front with your fingers at all. The oils from skin are the main cause of hazy, blurry video, especially at night when the IR light refracts off the smudges.