Note: below, full screen refers to the app full screen, not the player full screen.
I have two monitors, one 2K (main, external monitor) and one Full HD (laptop, secondary monitor).
I use Windows 10 with the main bar (the one with the "Start" menu) docked to the left of the monitor (instead of the usual bottom).
When I open Spotify, it automatically opens in full screen in the 2K, main monitor without issues; if I try to drag it to the second full HD monitor, however, it will be transferred to the new monitor still in full screen, but the right border exceeds the boundaries by a bunch of pixels (the close button on top right is almost entirely out of screen).
The issue can be solved by exiting full screen and then re-entering the full screen, all while being in the second, full HD monitor.
If I try to do the same actions starting in the 2K monitor (exit full screen, drag to the second monitor, enter full screen) the problem is present anyway; I need to (again) exit full screen in the full HD monitor and re-enter full screen in the same monitor to solve the issue,
Thanks for reaching out to the Community and welcome!
This can happen because the app might be dynamically resized based on the resolution that it thinks it's being run on and it sounds that either the graphic card or the operative system tries to eliminate any resolution differences between both screens, and it appears that this isn't synced with the app.
As this involves resolution switching and external hardware, we can't really provide a direct fix for it, as it really depends on the different setups and we can't optimize the app for individual usage scenarios and hardware, so the best way to go is exiting and re entering full screen, as you've been doing.
Hope this clears things up. If you need a hand with anything else, we're always a post away.
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