The most likely cause is dust on the sensor, not a separate video sensor. DSLR video and stills generally use the same imaging sensor.

Sensor dust is much easier to see at small apertures, and often becomes hard to notice at wider apertures. That explains why it may show up in video but not in your photo example: the camera was likely using a narrower aperture for video than for that still image.

To confirm it, take a photo in Aperture Priority and stop the lens down to a small aperture like f/22. If the spots appear, that strongly points to sensor dust.

Fix: have the sensor cleaned, or clean it properly using a safe camera-sensor cleaning method if you’re comfortable doing so.