The cork tenons are stuck onto plastic or wood. The small pieces on the keys are on metal. For the tenon cork, I'd suggest a simple wood glue. This is because when you change the tenons, you want to get the old cork off easily. If you use superglue here, it will be like cutting through concrete, and you can damage the clarinet.
For the key/lever cork, use something stronger. As Kflute said, yes, use superglue.
Oh, try not to glue your finger to the lever! It's so embarrassing at the hospital!
Steve