Hi.
Firstly, I sympathies to your woes. It's never fun buying a car then having major issues with it. Normally bearings fail almost instantly and it's down to zero oil or oil pressure, yes it's true there is an element of wear on all parts over a long period of time but with a constant supply of good oil it's almost impossible to fail like that.
There are of course some engines that have a poor design and do have a failure of bearings or other components but I'm not sure if this is the case with your engine.
The smoke can be a number of things, blue smoke is oil, black smoke is fuel, and white smoke is normally always coolant. Grey smoke is a mix of the above.
If your chap has skimmed your engine and by that I think you mean the cylinder head ? Then it's possible that coolant along with some other fluids may have escaped into your exhaust when your engine failed and what your seeing is the residue escaped from the exhaust system, this is almost always worse when it's cold (think of condensation) and will probably take a long while to clear quite possibly months.
However, I'm inclined to go back to coolant for this issue. Are you loosing coolant ? Check over the course of a week or two it has to be going somewhere. Your mechanic can also perform a coolant system pressure test to help expedite a diagnosis 👍
your injectors need servicing/changing for sure but I don't think it'll make a difference to your white smoke. But worth doing.
hope this helps.