Lets start with the actual model and spec number from the engine tag so we can see what setup you have. Also the model number info from the tag under the seat will allow us to see the rest of the system. Most likely that hose goes from a nipple on the air cleaner base to either the top of the fuel tank or to a charcoal canister.
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My model # is 917.288611
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I sent a pic of the engine tag
There should be a short hose attached to that nipple on the tranny that just points downward to act as a vent. Doesn't look like that hose have been in place for some time. There is also another hose that goes from a nipple on the air filter housing to a connector on the side of the engine. And then from that connector to a nipple on top of the fuel tank that captures the gas vapors and pulls then into the engine to be burnt.
Could you tell me where on the fuel tank the nipple goes? The nipple has broken off and I can’t tell where it was located. Will it run ok without this hose but do I need to repair it? Another issue I have is whenever I start it I have to use starter fluid to start it then it runs fine.
It looks like that fuel tank is under the seat so may have to remove the seat and rear fender assemblies to get to the top of the tank to replace the nipple if replaceable. The hard starting is most typically due to the choke not closing all the way. Put the throttle control in choke position and then go around to the front of the engine and see if the choke on the carb moves further. If so then just need to loosen the clamp on the throttle cable and side it to fully close the choke and then retighten the clamp.