How do I deal with the Rotten Vale?

Firing a Torch Pod at the ground will burn up the noxious gas, called effluvium, in a small area. Note, however, that only one area can be clear of effluvium at a time, firing a second Torch Pod will make the effluvium return around the first Torch Pod.

Alternatively, the armor skill Effluvial Expert makes you completely immune to the effects of the effluvium. It is found on the Hornetaur Mail and the Girros Coil.

For a temporary solution, wearing any mantle will stop the negative effects of the effluvium, presumably by filtering the air you're breathing.

Do note, however, that the effluvium itself does not cause the status effect that halves your maximum health. This status is inflicted by small monsters (Girros and Barnos) that are tainted by the effluvium, as signified by the rotten aura surrounding them.