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Currently, menu accessible for some users, which is invoked by desktop actions (for example by left-click on empty area in desktop) look outdated. There could exist favorites submenu related to current activity. I had some programs in favorites section (different for each activity) and need to fast access them.

Please be specific in what menu you are actually talking about. There is no default for such a menu. Certainly you can configure your desktop such that it opens a menu on left click, you then you have to tell us what menu that is.
All menues I know about can be configured and extended already. So most likely you already have the means to implement what you suggest. For a clear answer, again, you'd have to tell us what menu you refer to.

I think about programs menu/programs launcher. Sorry - maybe I give bad translation - in Polish it is called "uruchamiacz programów". I set it under left mouse button in desktop settings (under right click on desktop) -> mouse actions.