Pull the carb out, with the cable attached, and make sure the slide is actually closing.
Where is the hole? If it allows air into the motor when the throttle is shut, then it will be your problem. One of the many 'expoxy putty' products (Knead It, Metal Mend, etc) should be able to make a decent repair - but make sure the surface is spotless, and make sure that if the repair does fail that it won't suck the stuff into your motor...