Came across two tools that intend to bring the notifications from an android smartphone to the PC's desktop. This way, no notifications get missed. They are no replacement for e.g airdroid which brings the display of the smartphone onto the desktop and allows file transfers.
One is kdeconnect. I installed it, but only got an empty window when accessing it through KDE's control panel, so I couldn't use it at all.
The other one is linconnect and it works quite fine. It allows to determine whch apps shall get their notifications sent to the desktop. I recommend removing the mail apps (as usually a mail program is already running on the PC and informing about activity in the inbox) and also the GPS status apps (this one was updating the GPS status once per second, resulting in a flood of notifications)