The Core is an object that only appears in the Assault of the Dreadnought Gummi Missions. It doesn't attack, but it is often surrounded by various Nobody Ships and other enemies which appear in massive numbers. The Core drops a treasure in every mission if defeated.
What happens at the end of the mission is determined by whether or not the Core was destroyed. If the Core's shield appears before the player can destroy it, the Dreadnought fires a giant laser beam at the bottom of the screen. If the Core is destroyed, the Dreadnought explodes in a massive fireball. This sequence cannot be seen in Mission 3, as the Gummi Ship's attention becomes solely focused on the Hunter-X.