My Mini has hardy spicer uni joints. When I shake the drive shaft, there is play in output shaft and the seal. Is this caused by the nut being loose on the output shaft?

Or the bush in the side plate.

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1963 Morris 850 Smoke Grey
1963 Morris 850 Shadow Blue
1965 Morris Mini Smooth Van
1966 Morris Mini Van
1969 Morris Mini Deluxe Resto
5/72 Clubman Gt

or the bush in the diff housing / crown wheel?

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did I tell you that I won a trophy?

Can this be sorted with the engine in the car?

Side cover yes, diff bush no.

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1963 Morris 850 Smoke Grey
1963 Morris 850 Shadow Blue
1965 Morris Mini Smooth Van
1966 Morris Mini Van
1969 Morris Mini Deluxe Resto
5/72 Clubman Gt

Often you find the drive flange is worn unevenly, as well as the cover bush worn. It's unusual to find much wear on the diff cage bushes.
Replacement side cover bushes are all made undersize and need their bore machined after fitting.
If the drive flange is worn it can be turned down a few thou then the bush bored to suit. Allow 1-2 thou clearance.

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DrMini- 1970 wasaMatic 1360, Mk1S crank, 86.6HP (ATW) =~125 @ crank, 45 Dellorto (38 chokes), RE282 sprint cam, 1.5 rockers, 11.0:1 C/R.