The reason for the aux valve is to smooth out the idle when it is warm. They are fully closed when the engine is hot. If they stick in the closed position you will get a rough start and then, when the engine is warm and the aux valve should be closed, they smooth out.

I test these by first removing it from the engine. A bit of form-a-gasket or gasket sealer will be needed to replaced it. Or a new gasket.

Then bring a pan of water to a boil and drop the valve in. It should fully close as in 100% with no leaking of air from one port to the other.

The valve should be in the open position when cold. If it is in the closed position when you remove it then you know right away it is bad.

Anyway, take a small flashlight and, after using a pair of pliers to lift and hold the valve from the boiling water, shine the light into one port and look into the other. You should not see any light at all passing through.

You can also attach a hose to one port and try to blow through it. If you can blow any at all it needs help. You should not be able to blow any air through it when it is hot and sealed off.

An old trick is to pinch down the bulb in order to squeeze the rubber compound inside the valve. This rarely works well but it is worth a try.