When I try to do any animation (in cycles), it ends up with black specks. like a dirty screen that stays with the camera and moves over objects in the scene. I have tried many settings to fix this.
I think you are experiencing two things:
1: Noise due to a low number of samples.
2: A fixed noise pattern.
The fix for the first one, is increasing the number of samples.
For the second one, change the noise seed using the timer Icon. That will make each frame have a different noise pattern.