Wheel wobble usually results when the rim and/or tyre is running out of true, out of balance or not running concentric with the axle.
If the rim is running true, but the tread of the tyre is not, you may be able to let the air out of the tyre and adjust the seating of the tyre bead within the rim to get the tyre a bit more true to the rim.
If your rim runs true but your tread still visibly wobbles from side to side, it might improve your ride if you adjust the spokes to try and get the tread running as true as possible instead of the rim.
If you lift the wheel off the ground, and gradually apply the throttle, you should be able to see any noticeable runout of the rim/tyre.
If the wheel and tyre are running true at slow speed, but the whole wheel assembly is trying to move up and down as it rotates faster, this is more likely to be a balance problem.
Try placing a lump of plasticine or blu tack etc. against the rim
(by wrapping it around a spoke nipple) and see if the up and down movement improves or worsens.
keep moving it around the rim to different spokes until you find the best position that gives the least movement.
Now try adding a bit more weight, or removing some of the existing weight until you reduce the movement to a minimum.
With a bit of experimentation, it should be possible to eliminate the up and down movement completely.
Make a note of the position on the rim and then remove and weigh the temporary weight.
A local tyre depot should be able to supply you with some lead weights which have double sided sticky tape on the rear to allow it to be securely fixed onto the rim:
You simply break/cut off the amount of weight you require and then peel and stick it onto your rim in the previously noted position.
Spoke weights are more commonly used on spoked motorcycle wheels, but these would be too large for the much smaller spokes nipples used on pushbike wheels:
As your rims don't appear to have the lightening holes, it might be possible to place the weights inside the tyre so they are not seen:
Make sure that you mark the direction of the tyre, and its position in relation to the rim, so that it can be put back in exactly the same position, or it is likely to undo all the time you have already spent finding the correct location for the balance weights.
You simply place the rim tape over the top of the weights and the inflated inner tube will ensure that they will remain securely in place.
Alan