Win condition: Rout the enemy
If you’re playing on hard mode, there’s a good chance this map will be incredibly challenging. If you want any chance of saving Mathilda on Hard, you’re going to need to grind in advance. A lot. On normal, it’s more doable.
Just inside the castle there’s a rather large line of Arcanists, whom you’ll need to take out using Silque or your own Res-strong Mages as lead. Mathilda is quite durable and will last a while, but remember – if you retreat and return to battle, her chances will be much worse. The first instance of this battle is your best chance at saving her, despite its challenges. If you do retreat and return after defeating the majority of the Arcanists, Cavaliers will take their place the second time.
Desaix himself is a bit of a nuisance, but if you have Mages or Sages it will be a lot easier. Seraphim is a nice choice to obliterate him with, but otherwise simply cast damaging magic from a safe distance. The problem of course is that he’s so bulky that he can deal physical damage to even your strongest Knights, so remember to heal or switch units in and out of his path of destruction accordingly. If you can defeat him before Mathilda falls, you’ll have a solid chance at saving her.
Once inside the castle walls, get the Cantor as quickly as possible to stop the regeneration of terrors, and wipe out the terrors using Silque’s magic. Rout the enemy to clear the map, and if Mathilda survives then she’ll join you shortly.