The car also has a snow screen designed to keep out large debris and water out of the air intake box. It gets clogged with dirt and should also be cleaned. If it gets clogged, a secondary air intake flap opens and lets in warm engine bay air.
Since the air box cover is not in the way, this is also a good time to drain the fuel filter or replace it, see 1000q: fuel filter service for this procedure. This article also shows how to remove the airbox.
For the passat, remove the 4 metal clamps holding the top of the intake air box. For the Jetta, there are only 2 clamps. The air box top will then slide out.
Remove the top and replace the air filter. Make sure that it is seated properly
To clean the snow screen, unscrew the two screws and lift up. If the screws are rusted, use a dremel to cut a slot in the top and use a screwdriver to remove it. Also check the movement of the spring loaded secondary air intake flap - it should move freely.
Circled below are the locations of the 4 metal clips and the MAF plug.
Below are some more pictures of the airbox removed. The snorkel is an auxiliary air intake with a screen at the end. If the snowfilter screen gets clogged with leaves or a mouse nest, the spring loaded door at the back will open and let in warm air. If the engine didn't get any intake air it would stop running (stall).
To remove the airbox, remove the rubber band holding it down. On the Passat, remove both the auxiliary snorkel and main snorkel to twist the airbox out. On the Jetta, it's easier to remove the cold air intake snorkel to twist it out. Note the position of the rubber pegs.
For reference, here is a parts diagram listing of the parts for the Jetta intake system
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