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I don’t know what is going on with your system, but that isn’t normal. It shouldn’t even be minimised, the tray icon isn’t the UI but a means of accessing it.
When is this happening (only on reboot) and what are you doing at the time ?

I have three systems all with different OSes and three different versions of avast free (which I would have though it more likely to happen with if this was intentional) and none of them experience this issue.

With your reinstall, was this a clean reinstall also using the avast uninstall utility ?