White smoke, assuming it is diesel-smelling, is from bad combustion. Some of this is normal (it even says this in the manual, if anyone bothers to read it). But it is usually limited to a small "start up puff", and worse in cold weather. All of my TDIs do it, to some extent. Usually it is not enough to notice from the driver's seat, although the BHW does it enough below 30 F that you almost always notice it. It always has, since it was brand new.
But if you have large constant plumes, accompanied with misfires, then you have something else going on (keep in mind, a quarter million mile car after sitting outside overnight at 0F may do this no matter what).
If there is air (loss of prime) getting into the fuel system somehow, you can have this same sort of condition. Because the injection system cannot properly atomize the fuel if it is trying to compress a gas (air) at the same time as trying to squirt a liquid (diesel) under high pressure.
Loss of prime is usually easy enough to rule out. Because once the engine is running, and it has pushed the air out, it will typically hold that prime for at least a little while. So if as soon as it starts to run normally, and most of its start up smoke and fits are done (maybe ~15 seconds?), shut the engine back off and wait 10 minutes. Start it again. If it starts right up, and doesn't exhibit the same symptoms, it probably is a loss of fuel prime from sitting.
Keep in mind, on the PD, loss of fuel prime can actually happen internally in the engine... one or more of the injector seals could be weeping internally, allowing a loss of fuel prime in the fuel galley in the head itself (it doesn't take much volume... a tiny bit is all) and it leaks into the engine and mixes with the oil. If it is a small enough amount, you really won't ever know it, as it won't dilute the oil *that* much although it may contribute to some faster oil consumption.