A broken cannon is an item acquired by repairing Bill Teach's cannon aboard his ship The Adventurous during the quest Cabin Fever. To repair his broken cannon requires one cannon barrel (acquired from the gun locker below deck). Left-clicking the Broken cannon with the cannon barrel in your inventory automatically removes the Broken cannon to your Backpack and adds the new cannon barrel in its place. After repairing, the Broken cannon is useless and no longer required.
His cannon is initially damaged during the pirate attack, requiring you to repair it at least once during the quest. The cannon can be broken more times by failing to clean the cannon after each firing, resulting in an explosion and another Broken cannon.