Comment 1 for bug 1868524

Hi ,

For your errors

1-Dialogue window displays message: "No system tray detected on this system. Unable to start, exiting." - This will not effect the printing and scanning on your, this error can be ignored

2-HPLIP fails to install printer driver in Printers list in Gnome Settings dialogue. -
Add the printer by running "hp-setup" command in the terminal and follow the steps mentioned there.

3 -When I press 'print' or Ctrl + P the HP 1606dn fails to be listed in available printers-
When the previous setup completes the printer will now be listed.

4 -Checking for HP Properitery Plugin's.... No plug-in printers are configured. - This issue would seem to be at the heart of the issue!
Try installing the plugin from https://developers.hp.com/hp-linux-imaging-and-printing/plugins
and install it using
>> sh hplip-3.20.3-plugin.run