- The top of Mt.Fuji was covered with clouds.
- The top of Mt.Fuji was covered by clouds.
Both forms are grammatically valid, and both seem quite natural to me, a native speaker of AmE.
The "with" form would be more usual if the clouds were stationary, or came off the mountain as mist. The "by" form is more natural if the clouds re passing by the mountain. But these are nuances, and I can't think of a context in which one form would be acceptable and the other would be wrong.
I agree with the comment by FumbleFingers that:
Whoever told you by was "incorrect" in this context is an idiot ...