The hose on the very bottom of the carb (float bowl) is the over flow hose. the hose is open to the ground so if it the float bowl becomes over filled it will dump the excess fuel will go on the ground and not into the motor and or the air box. the device in line is a check valve to prevent any dirt and debris form coming up into the float bowl. that hose is also where the fuel comes from when the drain screw is opened. things that can cause the fuel to come out of that hose is a bad float, corrosion on the drain screw, needle and seat. dirt or some kind of debris in the needle in the seat area also improper float hight.
Test the floats needle and seat the carb will need to be removed from the ATV then take off the float bowl and turn the carb upside down. Hook the fuel line on to the fuel in let and allow the fuel to flow to the carb. If the needle and seat are sealing properly no fuel should come out of the needle and seat. If it dose flow from the seat area you will need parts to fix the leak.
to check the float hight can be done with the carb still on the unit. you will need a clear hose and put it on the bottom drain and open the drain see the attached photos.