I am also having this issue in 16.04, std/default desktop install.
Here's a work-around. Edit the ~/Desktop/xxxx.desktop file to remove the "URL" entry (e.g. change "URL=http..." to "xxURL=http..."). (Perhaps also need to change the "Type-=Link" line as well -- not sure.) Doing this made the desktop icon movable for me. Once you have the icon where you want it, undo the edits of the *.desktop file.
So it seems to be the presence of the URL entry in the *.desktop file which causes the problem.
I am also having this issue in 16.04, std/default desktop install.
Here's a work-around. Edit the ~/Desktop/ xxxx.desktop file to remove the "URL" entry (e.g. change "URL=http..." to "xxURL=http..."). (Perhaps also need to change the "Type-=Link" line as well -- not sure.) Doing this made the desktop icon movable for me. Once you have the icon where you want it, undo the edits of the *.desktop file.
So it seems to be the presence of the URL entry in the *.desktop file which causes the problem.