I always let the experts talk first but since no one has responded yet
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Go to the "Application Menu" and type "Login Screen (SDDM)". There, click on "Behavior...) is in the top center. You can pick even if Wayland or X11.
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Find Kate in the "Application Menu", right-click on it and select "Pin to task manager¨. You can easily drag it from there to the System tray.
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It is funny, right-click and Open with Kate does not bring it to the front, but double clicking it does fine... Never noticed that glitch.