First, that air cleaner element has a foam prefilter around the outside. You need to remove the prefilter and wash just the prefilter in petrol, then re-oil it with engine oil and put it back over the paper filter, each time you service the mower. You can only inspect the condition of the paper filter by removing the prefilter - people sometimes are fooled by a clean-looking prefilter and do not realise that the paper filter behind it is completely clogged up.
Second, I think you said that it runs properly with the air cleaner cover removed, but it won't run with the cover installed even if you leave the filter element on the bench when you do it, so you are running it effectively without the air cleaner. If that is really happening, the air cleaner cover has a blocked air intake. Please clarify exactly what causes it to not run: the air cleaner element, or the air cleaner cover.
Third, the mixture screw on the Honda/chonda carburetors is only an idle air bleed and has very little effect even at idle. It has no effect at all when the throttle is open. Furthermore if there were anything wrong with the carburetor, it would not run when you take the cover off the air cleaner. This is not a carburetor problem.
