That plastic tube is an overflow tube.
I'd pull the carb and tear it down.
First you'll need to give it a good internal cleaning, and make sure that all of the orifices are clean and clear. Blow thru them with compressed air to make sure they are clean.
Then make sure that the float is in good shape, also replace the float needle & seat. They're likely worn out.
Then once you have it all clean and the worn parts replaced, set the float height according to the service manual and you should be fine.