Removing Supports Inside Cone
That model didn’t need supports. Most of the time you can break the connections with the supports to the model by inserting a knife or flat object into the cavity. In this case you could use a probe to loosen the joins then wedge it out. If that doesn’t work you can use needle nose pliers to remove prices. 

Your best and cleanest option would be to reprint the mouthpiece without supports at all. Print it so that the wide section(the section touching your mouth) is on the buildplate.