According to the updated info in Tasha’s:
As a bonus action while the book is on your person, you can cause the mind to manifest as a Tiny spectral object, hovering in an unoccupied space of your choice within 60 feet of you.
As a bonus action, you can cause the spectral mind to hover up to 30 feet to an unoccupied space that you or it can see. It can pass through creatures but not objects.
At no point does it say the object can fly, but the only place I can find "hover" in the rules is in regards to Flying:
Fly A monster that has a flying speed can use all or part of its movement to fly. Some monsters have the ability to hover, which makes them hard to knock out of the air. Such a monster stops hovering when it dies.
…but again, the Manifest Mind is not given a fly speed. My thought is that the object hovers just above ground level, and does not have the ability to go above that. It’s also not affected by difficult terrain, etc. Any thoughts from the experts are appreciated.