Which TDI model do you own? A quart every 4K miles is slightly above normal for the A3/B4 models.
There can be other factors involved:
weight or type of oil: synthetic/non-synthetic, 10W-30, 5W-40 etc.
driving style: heavy foot/light foot?
oil leaks: CCV breather puck/crank seals/turbo seals/valve cover gasket etc.
As already mentioned,
plugged intake manifold: causes excessive smoke.
Plugged snow screen: same symptoms as a plugged intake manifold.
Excessively clogged injectors: will cause poor combustion and excessive smoke which will cause low compression which will cause excessive blowby which will cause excessive oil usage. (hows that for a run-on sentence
)
Check these areas over and get back to us.
IF these areas check out OK, then I would have a compression test or leak-down test done. This will tell you if it is valves, valve seats, valve guides, piston rings, and which cylinder has the problem.
DO NOT USE any oil additives until you know what is going on. Oil additives are only good if the damage is more than you are willing to spend to fix and you are only milking the car along until you are ready to dump it.
Here is some interesting information on
Lucas oil additives from
Bob the Oil Guy web site.
And one last bit to help complicate things: the oil usage problem and the smoking problem might be totally unrelated!
Good luck!