A wobbly clutch is obviously not good. Not only will it cause vibrations but it also will not allow the teeth of both gears to mesh together cleanly and quietly.
The noise you are hearing is most likely this bad gear meshing. That said, another notorious noise maker is a dry clutch bucking bar and bearing.
If you remove the sprocket cover you will see a round bar sticking out of the end of the sprocket shaft. That is the clutch bucking bar. Further up inside the hollow sprocket shaft, behind the bucking bar, is a single steel ball bearing.
The inside of the hollow shaft, the ball bearing and the bucking bar all need to be well coated with grease at all times.
The last somewhat common cause of noise are the sprocket shaft bearings. If they do not have enough grease inside them they can howl like a banshee.
On page 2, post #12, of Bradwood's thread "Heavy drag when pedaling"", MarkSumpter shares an excellent way of greasing the transmission shaft bearings. It also ensures that the internal clutch spring assembly is adequately greased as well.
Click here for MarkSumpter's post
FWIW, you have waaay too much wobble in the clutch. The question is... is it a bad clutch or a bad shaft?