Student or Learner
Their eyes met and something passed between them before Dad continued. “Your semester here at Hecate Hall is almost over. Come spend the summer with me, and at the end of that time, if you still want to go through the Removal, I’ll allow it.”
My eyebrows shot up. “What, like at your house? In England?” My pulse sped up. There had been three sightings of Archer in England.
three sightings ??
Archer had been spotted three times in England.
We normally tend to use "sighting" for things like criminals on the run, people wanted by the police and so on. When they have been spotted somewhere it's often referred to as a "sighting", especially if it's being reported on the news somewhere. It's quite an idiomatic use.
It can be used for other things, too, but I think that's the sense you want here.
It also gets used for things like "There have been only four recording sightings of the rare Osprey in the last year."