Pedals wobble and click under load
Examine the pedal and crank arm.

(Note that the above diagram is not exactly like your pedal, but the basic construction is the same.)
Note that there is a shaft that runs inside the pedal, from crank arm to the end of the pedal.  The pedal bearings (inside the central housing of the pedal) ride on the pedal shaft.  Likely the bearings have gotten loose and the pedal is rattling around on the shaft.  If this is the case you should go ahead and replace the pedals.
But carefully examine the shaft as you move the pedal up and down.  Where the shaft mates with the crank arm you will be able to see 10-15mm of the shaft, and you should not see ANY motion between this shaft and the crank arm.  If you see any motion there (while you move the pedal up and down) it means that the pedal is unscrewing from the crank arm and the crank arm will be very quickly damaged (if it isn't already).
Further, grasp the crank arm and move it in and out, checking for any flex between the arm and the crank shaft inside the bottom-bracket.  Any motion here must be corrected immediately (tighten the crank nut) or the crank arm and possibly crank shaft will be damaged.