Are you talking about the icon with two computers sitting in the panel down by the date & time in the default installation? That is the nm-applet (Network Manager) for your system. If that is the Wired Network Connection icon, I am unsure if you want to do to much fiddling with that. It might lead to some future problems. I could be wrong, but I would be wary of wholesale deletion at this point.
What you are describing ( mintMenu > Preferences > mintDesktop ... Desktop Items ... Network ) is not the same as the nm-applet in the panel. They are different. Please make sure of what you need before pressing the delete key. Maybe I am just over-cautious.