Land Rover Repair: 1997 Discovery Land Rover, discovery land rover, ventilation system

Question
When the a/c runs for a while a mist of smoke comes from the vents. I had the rover check out and the mechanic said I needed a heater core and a water pump. Does this sound right.

Answer
No, it doesn't. Are you having to add coolant?  If so, it is possible it could be the heater core and the water pump, but these are separate issues.  The heater core would cause steam to enter the cabin, whether A/C or heat was selected.  Also, if the water pump was leaking that badly, steam would enter through the ventilation system, whether heat or A/C was selected.  But with the water pump, for steam to enter the cabin, I would think your engine would be nearly destroyed.  

If your heater core is gone you will have a slimy film on the interior glass, throughout the car.  When you wipe it, it smears.

I doubt you are actually seeing smoke.  It is more likely vapor caused by overchilled evaporator or blocked A/C drains.