This bug is present under X and Wayland alike.

Start KDE Kontact. Reduce it by minimizing. I will show in the taskbar as
running program and you can reopen it. 
Now from open, close Kontact window, by clicking on the outer x of the window.

Hence here we CLOSE the window but we do NOT exit the application. 

Normal behaviour wants that if you click on the icon in the tray, the program
will reopen, whilst the tray icon allows also to call for certain actions. Why
it seams possible to execute "search for mail" and alike, a click on the tray
icon does not result any more as before in an opening of the application. Hence
you have to go to the start menu, search for it, start it and then only you can
use it. 

Observed behaviour: click on icon does not open program any more
Expected behaviour: click on icon restores the window of the application
running in background 

PS. I do not think the issue is of Kontact, but of Plasma therefore I filed the
bug against it. Feel free to correct this when assigning.

Kontact version 6.0.1 (24.02.1)

Operating System: openSUSE Tumbleweed 20240329
KDE Plasma Version: 6.0.3
KDE Frameworks Version: 6.0.0
Qt Version: 6.6.3
Kernel Version: 6.8.1-1-default (64-bit)
Graphics Platform: X11
Processors: 12 × AMD Ryzen 5 5600G with Radeon Graphics
Memory: 62.2 GiB of RAM
Graphics Processor: AMD Radeon Pro W5500
Product Name: X570 Phantom Gaming 4