This is the inlet pipe for the air filter. It is most useful in colder weather, when it is moved to draw in air from near the hot exhaust pipe, making the intake air a bit warmer. It looks like in your picture the hose is in the 'cold weather' position. In warm weather, the end of the pipe furthest away from the air filter is removed from near the exhaust pipe and clipped to the inner wing to draw in cooler air.
So it's not essential to have this pipe, but it make help the car to run better in cold weather.
In fact, it looks like in your picture your air filter has two inlet hoses: the flexible hose (with the red arrow) and a shorter solid, black metal pipe further round to the left of it. This second pipe is also pointing towards the exhaust pipe, so will be drawing in warmer air. I think that pipe can also be moved to draw in cooler air.