Tales of Vesperia: Definitive Edition

During the fight with the Outbreaker, it will periodically spit out its core and hold it in front of its face, then suck it back into itself. It's pretty obvious when it does it because it will completely stop attacking and moving. Also, it will continue spitting it out periodically the whole fight until the core is destroyed, so you don't have to worry about boss HP cutoffs or anything like that.

The objective of the secret mission is to destroy the core using aerial attacks and Artes.

Using Judith or Yuri makes this pretty easy. Yuri's Severing Fang is by far the easiest way in my honest opinion to get the mission. The Arte hits the orb multiple times each time Outbreaker spits it out.

Another great Arte to use is Rita's Tractor Beam. The timing is a bit tricky, but sometimes the boss will spit out his core back to back. If you start casting right as you see him spit out the first time, by the time he spits out the second time (if he does two in a row) the spell will go exactly as he spits it out!

Just keep whittling away at the orb until you see Great!!!. Set everyone to Defend if you're worried about killing the boss. You can let the boss absorb the core so it will heal itself if you need more time to kill the core.