Looks like it goes to the water pump. Perhaps a heater bypass hose. I wouldn't replace it until you fix the oil leak or the new hose will rot too.
Edit: Cancel that. I think "Vent Tube" is correct. :thumbsup:
I don't think the hose is rotten, it has just split towards the bottom end where it starts to slide over the metal stem.
What's the prupose of the vent tube?
If it is the 'vent tube' what part is it called ? Or part number etc so I can order one?