Is there any way to remove it without losing icons on the desktop?
I think this is a bug. If you could move the shell's panel, you'd see that menu goes right across the entire screen and provides File/Edit/View/Go/Bookmarks/Help.
It's caused by the hacks Unity has to perform to get a menu for the desktop. Yes, I know you're not using Unity. Neither am I (Gnome Classic) and I've got it too.
Unfortunately the only way it's going to get fixed is if somebody improves the way that Unity deals with the menus, or they elaborate on the hack so it only does it when Unity is running.
There are patches floating around (check out the bug report) but I couldn't say if or when it's going to get pulled in.
Have you tried to use the gnome-tweak-tool and turn off the file manager handing the desktop as described in this solution to a similar problem: