I too like my desktop clutter free, so I don't have any desktop icons 'shortcuts' displayed on my desktop. I use RocketDock for easy access for commonly used Programs, Functions, Explorer, frequently used folders, etc.
Even for the most part I don't allow programs to set a desktop icon were there is an option to allow/deny it. I don't believe avast has an option, even in the custom installation not to set a desktop icon.
The only advice I can give is don't delete it as it will just keep coming back or in your case with so few desktop icons hide them all.