Mower engines run on the idle jet until you engage the blades or start driving up a hill
That is why the engine speeds are called "low Idle " & "high idle"
So you have a problem with the main jet or the fuel supply to the float bowl.
The tube you are asking about is the tank vent.
EPA regulations now require the tanks to be closed to the atmosphere so no fuel vapour can escape which of course will cause all life on the planet to end.
The stupidity of this is these systems failing cause more fuel to be spilled than what would evaporate out of a tank by a factor of probably around a thousand if not more .
Same story with you float bowl vent being on the engine side of the carb so you do not know the float valve has failed till smoke pours out of the exhaust and the sump has 1 qt of oil & 3 qts of petrol in it most of which will end up evaporating because no home owner has liquid volatile waste disposal gear, in fact very few workshops have them either .
So I tip the petrol diluted oil into my waste oil drum and eventually it all evaporates .