The Hunting Quest is a series of repeatable quests where you are pitted with certain type of enemies in order to farm certain Ascension Materials. There are three type of difficulties from easiest to most difficult, each with different AP Cost and a few Mana Prisms as first clear rewards.
First Clear Reward
These will use the Pride difficulty as the basis.
There's only one type of enemies and they're not bosses, so bring a party of Servants with class advantage to clear faster. Be sure that they're in high levels beforehand.