In previous versions you could move icons simply dragging them around in the favourites menu. On the contrary, in recent versions the icons are unmovable.

The first workaround is to rearrange them from
~/.config/mate-menu/applications.list

The real solution is to fix the bug in the python plugin.

In my case Mate is on Debian 11, however this path should be the same almost in every distro.

sudo nano /usr/lib/python3/dist-packages/mate_menu/plugins/applications.py

~LINE 191

class pluginclass( object ):
    TARGET_TYPE_TEXT = 80
    toButton = ( Gtk.TargetEntry.new( "text/uri-list", 0, TARGET_TYPE_TEXT ), Gtk.TargetEntry.new( "text/uri-list", 0, TARGET_TYPE_TEXT ) )
    TARGET_TYPE_FAV = 81
 
    <del>#toFav = ( Gtk.TargetEntry.new( "FAVORITES", Gtk.TargetFlags.SAME_APP, 81 ), Gtk.TargetEntry.new( "text/plain", 0, 100 ), Gtk.TargetEntry.new( "text/uri-list", 0, 101 ) ) 
    #fromFav = ( Gtk.TargetEntry.new( "FAVORITES", Gtk.TargetFlags.SAME_APP, 81 ), Gtk.TargetEntry.new( "FAVORITES", Gtk.TargetFlags.SAME_APP, 81 ) )</del>
 
    # Rik
    toFav = (Gtk.TargetEntry.new("text/plain", Gtk.TargetFlags.SAME_APP, 81), Gtk.TargetEntry.new("text/plain", 0, 100), Gtk.TargetEntry.new("text/uri-list", 0, 101))
    fromFav = (Gtk.TargetEntry.new("text/plain", Gtk.TargetFlags.SAME_APP, 81),  Gtk.TargetEntry.new("text/plain", Gtk.TargetFlags.SAME_APP, 81))

Line 1371

def on_drag_data_received( self, widget, context, x, y, selection, info, time ):
        if info == self.TARGET_TYPE_FAV:
            <del>#self.favoritesReorder( int(selection.get_data()), widget.position )</del>
            #Rik
            self.favoritesReorder(int(selection.get_data().decode()), widget.position)
 
def on_drag_data_get( self, widget, context, selection, targetType, time ):
        if targetType == self.TARGET_TYPE_FAV:
            <del>#selection.set(Gdk.SELECTION_CLIPBOARD, 8, str(widget.position))</del>
            #Rik
            selection.set_text(str(widget.position), -1)

Save, reboot and the problem is solved.