As Andy said, you need to find out where the play is coming from.

Jack up the back end and heave the wheel from side to side. You should be able to see where the play is coming from.

It is most likely to be the ball joints in the toe link. (The locating arm in your top photo) but it could be wheel bearings or play in the bottom arm front bushes.

Whatever it is you should be able to see the movement when you heave on the wheel.

This may just be me being thick:

If you've got a tie strap like that for locking the steering, you shouldn't have a-arm bottom "wishbone"

No.
You use the standard Mini top arm, bottom arm and tie rod as normal then just lock the steering with the fixed toe link replacing the normal steering rack.

So i'm right then???

Anyway, all your bushes for those bits are easily available from any mini specialist.

If it's the tie-bar, then it's Rossa bitz.

Hooray...! I put it all back together with the new C washer and it appears to be working.
Thanks to everyone for your help and encouragement.

Ed

Well done.

Does that mean it is back on the road?

Sadly not. I have loads of other jobs to do, but this was a big one.
My next obstacle, aside from the ball joint is the tuning. In thoery Once those two things are sorted I could book it in for an MOT and see what it fails on... But it is getting close now. I've had the car for nearly a year now and still haven't driven it, which is frustrating. I am tempted to put the door back on and drive it round the block- a bit naughty though, so will probably just do a bit of to and fro-ing in the drive.

Ed