Have just found the resolution, entirely by accident.

On my laptop, the issue appears to be caused by allowing the "ATi Catalyst Control Center" to display the associated system tray icon. With that tray icon disabled, the small white box in the upper left corner of the desktop does disappear...and with it enabled, the box immediately reappears.

Checking the system logs, this driver was indeed recently updated, and I did first notice this trouble at a time after that update.