If by the black sprites you mean the enemies then they are actually from the academy, as far as I know at least one of the zones is a spiral that you access from one of the classroooms, it also leads to anothere classroom, I am exploring the desert now but I assume the other zone is also connected to the academy for lore reasons and because those are the enemies that appear there, this game does not tell you where to go, the desert apparently comes first.
By the way there is a secret passage in the garden at the castle that allows you to skip the pure material requirements, it is on the left fountain, it leads to a prison of sorts and the stairs there lead to the different section of the skill room, the loked cell can be accesed from the roof, the stairs on bottom left cell lead there, it is just a memory but it could be usefull to know.