I have always seen the Imperial Nobility as being far more concerned as to running the space between the stars than what occurs on world. They protect trade, ensure the worlds security from outside threats, and keep things moving between systems. The spice must flow.
An Imperial Noble's only duty on a world are to maintain the few Imperial ground side facilities (including but not limited to his fief and the starport), ensure that the world follows the few Imperial rules and pays it's taxes on time. The 3rd Imperium canonically leaves the governing of an individual world to the individual world. This is why there are so many Dictators in possession of worlds and why there are both planetary nobles AND Imperial nobles.
To be quite honest, I don't see a world's government as even having any say in the Moot at all, as the Moot does little in the way of Governing. Its only powers are to confirm a new Emperor and to dissolve the Imperium. Neither of which is any concern of a world government. Aside from having to keep a few 3I rules in place, a worlds government is not Imperial, and the Imperial government is not the worlds government.
I do see the Worlds Imperial Noble acting as a liaison in multi world disputes. The Nobles of the various worlds in question acting as impartial diplomats and go-betweens, but not much else. I also see the Noble being the guy to whom a worlds government turns to when they want Imperial intervention, intercession or assistance. But not much else.
As far as a planetary government is concerned, an Imperial noble is both strong and weak. Weak in that they have NO say in what happens outside of the worlds Imperial facility's or concerning the few 3I laws that must be kept. And yet they are strong in that they can cut off all trade with other worlds, have the worlds government removed or even have the world scorched black by the fleet should there be violations of Imperial Law.
Simply put, an Imperial Noble is associated with a particular world, because that world is the major world(s) in the territory of space that he has been assigned by the Emperor of the Third Imperium. He is both a power unto himself on the world and at the same time powerful and powerless where the local government is concerned.
Somehow I doubt my above ramblings have cleared up anything for anyone. But that's my .02CrImp