Are you saying that you download an app to, say, the Desktop, and when you drag it to the AppFolder it copies the thing rather than moving it, in other words you still have a copy on the Desktop? That is seriously weird in of itself, and needs examining.
But in the meantime, try Get Info on Activity Monitor, at the bottom where is says 'Details" click on the triangle and see what the permissions are, and whether the file is locked, the golden padlock. Change permissions to Read Write and if padlock closed click on it to open it. Also, if you did a straight update, go Apples support site and download the standalone combo updater, and apply, this often fixes such things, and if you did that already, reapply the combo another time.