Express as clearly as possible

I don't think there is one word that means "express as clearly as possible". That phrase reads OK and there is no obvious reason to change it.

The punctuation is not quite right. You should delete the comma after "that" and add a comma after "above-mentioned class".

"thought atmosphere" is a bit odd. I think you should find another word there. Possibly "mood"?

by thought atmosphere I meant the prevailing thought system of the corresponding class

that is likely to have an effect on people

The words "express as clearly as possible" are fine as they are.

I concur with others that "prevailing thought atmosphere" does not make sense. Perhaps "prevailing mindset" would work.

From American Heritage Dictionary: mindset - A fixed mental attitude or disposition that predetermines a person's responses to and interpretations of situations

So are these statements right?

However, when this primary use gives way to a secondary use, manifests themindset, beliefs, values, or, in other words, the prevailing culture of the society.

Furthermore, we can maintain that, the painter was trying to delineate the views and beliefs of the above-mentioned class and by juxtaposing these symbols in this way, his intent was to expr

Or "attitude" perhaps?

soheil1Furthermore, we can maintain that, the painter was trying to delineate the views and beliefs of the above-mentioned class and by juxtaposing these symbols in this way, his intent was to express as clearly as possible the prevailing mindset of this class.

I know it's easy enough to do when copying and pasting stuff, but at least twice now you have re-ask

Oh sorry.
Why is the comma necessary after the above-mentioned class?

soheil1Why is the comma necessary after the above-mentioned class?

It's quite a long sentence, and that is the start of a new clause, so it just helps to make the sentence more readable.