I would wish that the "HP Support Assistant" would be respectful to the choices that I make.
Please understand that when I choose to remove the HP Support Assistant toolbar through the settings for the taskbar I do that because I do not want the software to be in the taskbar. If I would have wanted the toolbar to be shown in the taskbar I would not be removing it (through the settings)
Windows have a place for background software and their notifications to be at , that place is known as "notification area".
There are also a function to show messages already, the icon for that i at the end of the taskbar.
I know that I can remove the notifications through the settings in HP Support Assistant so that I will not have the toolbar to re-appear and that makes me not to be notified on updates ... I want the notifications but not the toolbar.