Stationary bike "clicks" when pedaling - crank axle is moving sideways. How can I fix this? I've got this stationary bike, a York C202, which has developed a problem - namely it "clicks" when pedaling. This is highly annoying, as there's nothing I seem to be able to do about it.

I diagnosed the fault as per: the pedal crank axle moves sideways by about 1mm every revolution of the cranks.

I have no idea how to fix it though. I was googling around for different types of bottom brackets, but this bike seems to use some custom one.

There's a nice diagram of all the parts involved on the last pages of the PDF manual.

The C Clips are inserted into grooves on the axle itself, so I can't move them - it seems logical that it should be possible to either move the bearings further apart (screw them out of the frame by 1mm) or put some "placeholder" between the C Clip and the bearing. 

However - the C Clip rotates against the bearing, so it wouldn't be a secure fit for any contraption.

Can you advise? How can I fix this? I imagine I'd need some special tools for the job, which ones will I need?
The bike has individual cartridge bearings -- not at all standard for a "real" bike, and a bit unusual for even an exercise bike.  (In fact, though I've seen a half-dozen different types of exercise bikes taken apart I've never seen anything like this.)

But it's not a terribly complicated setup.  The C-clips hold the axle in place.  The ring against which the C-clips ride is the inner race of the bearing and rotates with the axle.  If you can find a bushing (washer) of a suitable diameter and thickness it can be inserted between the C-clip and the bearing -- fairly easily accomplished by someone with moderate mechanical abilities.  You should only need the bushing on one side.  The one thing to be wary of is to get the crank arm back on tight, as a loose crank arm quickly destroys the mating surfaces.

A well-equipped hardware store should have a selection of bushings (and some spare C-clips, should the one on there be damaged or go flying across the room).

(There is also an old mechanic's trick:  You could work some coarse string into the gap, twisting it around the axle several times.  It should stay in place fairly well, without anything to secure it, especially if you dab on a little grease or petrolatum.)