Bicycle Tutor

I have a pair of platform pedals I use on my commuting bike. When I started hearing a clicking noise while pedaling, I checked the usual suspects. I ended up with one pedal that is loose on the spindle. It will slide side to side. I can take a nut off under the dust cover on the outside and there is a thick washer under it. But I cannot tighten it enough to take the slack out. The other pedal is nice and tight in this direction. Both pedals spin just fine.
I don't see how to take the pedal apart farther. Any ideas for working on this pedal rather than just replacing them?

Usually there are 2 nuts separated by the washer. Remove the first nut and washer and see if there is another underneath?
The second one is for adjustment with the first one being a locknut. It can be a bit fiddly to tighten the locknut without the adjustment nut moving so be patient. Obviously too slack and the pedal wobbles or to tight and you get drag on the pedal.
If you really feel up to it you could completely strip the pedal and inspect the bearings. Clean, re-grease and re-assemble. N.B. take a note of the number of bearing balls at the front and back as they may be different. (possibly more at the back?)
Cheap pedals may have plain bearings but can still be cleaned and greased.