A broken cannon can be found by using a cannon barrel (from the gun locker) on Bill Teach's cannon. You then receive the item in your inventory. It is then useless and no longer required.
Bill Teach • Pirate • Mama
Fuse • Repair plank • Tacks • Plunder • Cannon barrel • Canister • Gunpowder • Cannon ball • Ramrod • Broken cannon • Book o' piracy
Cabin Fever • In the Brine