Dont know what cog you are talking about, do you mean the front sprocket? Thats supposed to be a bit wobbly.
Your chain adjuster is a bit different from my bike but you have to move those nuts on the thread just above the axle on the swingarm, this moves the axle back and forth, when you move it forward to loosen the chain hit the wheel with a rubber or wooden mallet to get it to go forwards.
Your chain looks like its at an allright tension, maybe a little bit loose..