The hose is for a battery gas vent. There is a chance that the leak is not at the battery door....but, it could be above or around the place it hows. I have seen window eyebrows installed above the battery doors too. Just chase it down.
"If it can't be reduced, reused, repaired, rebuilt, refurbished, refinished, resold, recycled or composted
then it should be restricted, redesigned or removed from production."