thoughts..
There should never be any axial movement, unless a force is applied that exceeds that which is imposed by the wavy washer. That is true whether the bearings are press fit onto the spindle, like Campy's or whether they are press fit into the cup and allowed to slip over the spindle. If the bearings are press fit into the cups, any fretting wear from axial movement will damage the expensive and nonreplaceable spindle, instead of a cheap bearing cup. If there is any real freeplay, then the wavy washer is shot or the BB is too short.
The amount if movement that is allowed in the UT system is limited by the spring clip on the right side. It should be only about .5mm. If you really think that is a source of a problem, shims could be installed between the spring clip and right bearing.
If you have worn bearing cups or damaged bearings, cleaning and regreasing won't fix that. Worn bearing cups can only be identified by measuring the ID, or at least looking for obvious signs of wear. The nickel plated cups might have the plating worn through, for example. In that case, spend $20 and get new cups.
Clicks could also come from simple problems like needing to lube the contact areas between the chainrings and crankarm spider and retighten the chainring mounting bolts.
Clicks and cup wear can also be the result of improper cup installation. The Campy instructions are not very good and may lead to problems. With new cups, I would always insure that the BB faces are square to the threads first, then grease the BB shell threads and torque the cups to spec rather than use loctite.
Misusing loctite is another potential problem. If the cups are new, they have a large amount of thread locking material on the threads. You can't leave that stuff in the threads and apply loctite over it. For one thing, it should be impossible to hand tighten the cups with the facotry thread locker in place. To use the loctite method, the factory apllied material should be remove completely before any loctite is applied. The loctite instructions clearly state that all threads must be clean, dry and free of foreign material before applying loctite.