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I have a single phase lathe that has a smoking/burning capacitor. The capacitor is 16uf 450v cbb60. Is this the correct capacitor for single phase @240v as it looks like it has been retrofitted ? Another capacitor that is connected to the machine is 220v 75uf. This one stays cool.

The capacitors (1 or 2) depend very much on the motor design. I just checked a motor starting capacitor which I salvaged from somewhere and it is 15 mfd 300 vac CBB65A, pretty close to yours so I think probably the one which is smoking should simply be replaced with a similar one. Single phase induction motors commonly require a starting capacitor which is only used for a second or less when you start the motor and is disconnected by a centrifugal switch on the motor shaft once the motor is spinning. It would be wise to check that this switch is still working and to check the starting winding for signs of overheating. Some motors have a second capacitor which is in circuit all the time connected to the main winding.

The main concern I have is the fact that the capacitor is rated for 450v and seen to have been replaced before. The other is rated to 220v and seems original. The micro motor is a two-valve capacitor motor.

Correction: I went back to the books (65-70 years ago) and found that 16 uf would be a running capacitor, not starting and is connected permanently to the starting winding not the main winding. Beyond that, I would still recommend checking the starting winding before replacing the capacitor. Regarding voltage rating, for a capacitor, as long as its voltage rating is more than the applied voltage it's ok. However, you have to be careful interpreting the marked voltage rating. If it says "vac" you can use it for any voltage up to its marked rating. If it says "vdc" you must divide it by 1.414 to get the ac rating (assuming a sine wave). This is because the ratio of peak to rms voltage of a sine wave is 1.414, (square root of 2). So in your case, rated 450v, if it's a dc rating you can use it up to 318vac well above 240v.

Ok cheers for that. Is there an explanation as to why the capacitor would have started burning up while running ?(with this being the running capacitor). And now, the motor will start but not run at speed but the cap just smokes until power trips. Would this not suggest that the cap is at fault ?

Certainly the capacitor is at fault if it's smoking and it must be replaced. The question is "What's damaging it?" It could be a random failure and in that case the only answer is to replace it and see what happens. Another scenario floating around in my mind is that when the centrifugal switch opens the starting winding generates a high voltage spike which damages the running capacitor. This would be a design fault and should have been caught in prototype testing but perhaps cost reduction was pushed too far. In any case the answer might be a MOV (metal oxide varister) connected to the capacitor terminals. Its voltage rating would have to be below the capacitor voltage rating but if too low it will explode like a firecracker