Pull the carb and see what critters have moved in to live. Check the diaphragm for tears or holes and give the carb a GOOD cleaning don't just show it the carb cleaner. If you have an ultrasonic cleaner run all of the metal, not the diaphragm through that. If run a wire through jets and other orifices on the carb and it it all cleans up well just get a new cap. Otherwise get a new carb.
IMHO, since the cap is cracked, the vacuum slide diaphragm has been exposed to the elements and may be dried out or have physical damage. A new one from SPC is $40.
Don't know about sourcing the cap, but there are a lot of CVK 30s out there. It's a pretty common small displacement motorcycle carburetor.
You might be able to repair that hole if there is enough material left to make a patch from fiberglas or JB Weld/epoxy and a piece of sheet metal or hard plastic.
I can get the cap easily from dealerships here or Italy I think that's my best bet. It starts but bogs out quickly, ill empty the tank, give this carb a good seeing too and patch up that hole with some fibreglass. Also cheers I didn't know what carb the 2004 had that will be helpful.
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