I took the back wheel off and noticed that when I hold the axle there is s amll bit of movement/play in it?
Is there a way to tighten this up or does it mean a new wheel?

Are you having play back and forth along the length of the axle? Most likely you need to tighten the cones, which are the circular things on the inside of the hub. You either need a really thin wrench and a normal one, or buy some cone specific ones, or your LBS can fix this really quickly. Illustrations always help if you're still not sure.

edit: If you have quick release then that's not a problem. It's supposed to be a bit loose when not engaged.

Can be a simple matter of adjusting the cone and locknut on the axle but you did not indicate what kind of hub/bearing/axle etc you have so it does depend on your actual hardware.

Kind of like telling the doctor "I've got an sore finger - do I need a new arm?" i.e. much more information needed to be helpful.

With quick release systems, the QR will compress the axle. This takes out a minute amount of slack so people usually leave a tiny amount of slack in the cones. Try feeling for the looseness when mounted on your bike.

+1. With the wheel installed, try to displace the wheel side-to-side with your fingers. If there is no play, then you are good to go. In fact, this test is the most accurate since a) it is in the as-used condition and b) any play in the hub is highly exaggerated at the distance of the rim.

Yes, it's secured with nuts.
When installed, I shake the wheel and it moves a bit side to side.
Do I need to take the freewheel off to do this?
Is it a case of just tightening them against each other?

I'll guess you have a standard cup and cone system. See this site for instructions on how to adjust (scroll down to maybe the last 3/4 of the page):

https://www.parktool.com/blog/repair-...and-adjustment

You'll probably need a cone wrench (flatter than usual mechanic wrenches) that fits your cones, and a regular mechanic wrench for the locknuts. It's an easy procedure.

Gordon: It is NOT just a matter of tightening them against each other. Please follow the Sheldon Brown or Park Tool instructions above or get some assistance. It is not a difficult adjustment to make but you need the proper tools and a little patience. Getting it wrong can result in damaged parts.

No, you don't need to remove the freewheel, you can just adjust the other side.

You do not tighten the right cone against the left cone. Rather, you tighten each cone against its adjoining locknut. In your case, leave the freewheel cone and locknut alone. You may have to remove a rubber dust cover on the left cone and locknut. Basically, hold the cone with cone wrench, use second wrench to loosen locknut. Operate the cone just to remove any play in the axle. Then hold the cone position with the cone wrench and turn only the locknut to the cone and tighten the locknut against the cone (make sure you keep the cone at the same position throughout the operation).

You may have to repeat a few times to get the hang of it. There is a sweet spot where there is no play with minimum drag on the bearing. Once you lock it all down, spin the axle with your fingers or hold the axle with your fingers and spin the wheel. Spin should be buttery smooth with no play in the axle. For fun, you may want to over tighten the axle and spin it just to feel what an incorrectly tightened system feels like.