I removed the stabilisers from my child's bike, but now when she rides there is a clicking noise and you can noticeably see the rear wheel wobble slightly, and the chain comes off frequently because if it.(wheel moves horizontally not vertically). I tried tightening the nuts but then it makes the wheel very hard to spin/turn. I'm assuming it is now requiring something to hold the axel n place
The chain drops because the chain's tension is too loose when the wheel was reinstalled. But that's secondary - first is to get the axle spinning right.
You've managed to loosen the cone nuts that form part of the bearing surface for the rear wheel. Then by tightening them the bearings can't roll smoothly.
The fix is hard to explain in words - lets try. Start by getting the wheel out of the bike by loosening the two outside nuts, slide wheel forward, flip the chain off the cog, and then slide wheel out the back.
There are three nuts on each side of the hub - the big one on the outside that holds the wheel to the frame dropouts, and then two more on the inside of the frame. The thin one is called the locknut and the innermost one is called a cone nut
First, figure out which side of the wheel has loose nuts. We need to tighten the cone nut to the point the axle has minimal perceptible movement sideways, but is still free to rotate. Too loose and it clangs around like a bell-clapper. Too tight and it can't rotate smoothly. Need to find that sweet-spot.
Then once you've found it, the cone nut needs to be held still while the locknut is tightened down on top of it. These two nuts together can't move in or out (hence locknut) The proper tool for this job is a thin spanner called a cone spanner, looking like this:
As you can see they're super-thin and often made out of pressed steel. They're quite specialist, but can be made if you're handy, or bought cheaply from on-line sources. A single 13/14/15/16mm one could be a few dollars.
Another photo showing how to use a cone spanner in cooperation with a regular spanner on the locknut.
This all sounds complex but its really not - more dirty than complex. Someone handy with bikes can fix this in a few minutes, less than it took to write this up.
The axle is held in place by the nuts that are visible in the picture. You should be able to tighten them so that the wheel does not move without stopping them from rotating.
From the description it sounds like your hub has loose lock nuts and attempting to tighten the wheel nuts interferes with bearing adjustment. You need to adjust the bearings and tighten the lock nuts before installing the wheel. One set of instructions for that can be found here. If this looks complicated, any repair shop should be able to do it, and if you bought the bike at a local store it should be covered by warranty.