Re: preposition

You want to know why 'at X desk' in one context but not the other?

For a start, in question 1), 'on' is not given as an option anyway.

At or on could both be appropriate, but mean different things, regarding desks anyway.

At the desk - this is using 'desk' as a location not a piece of furniture. The reception desk of a building will have a staff member who is actually performing the function, not the desk itself. You can register at the desk (but the person is doing it, not the desk), make a phone call from the reception desk (of course, you are calling from a phone, not the actual desk) or leave a package at the reception desk (in charge of the person, not just randomly abandoning it with a piece of furniture). In the process of leaving the package 'at the desk' the messenger may indeed place the package 'on the desk' but that is not what is meant overall. Leave at the reception desk = leave it at that recognised location (reception area). It is not a specific person's desk, or a specific person, the location is the important thing.

On the desk - this means to physically place something on the desk (as opposed to the floor, or a chair). Presumably the person giving the instruction does not expect to be there when you do so, it is an instruction to just abandon the report on top of the furniture. They could have asked you to leave it 'with my assistant' instead. It wouldn't be 'at my assistant' because the assistant is a person not a specific location. This time the desk itself does not count as a recognised location - needing 'at' - as it is the owner of the desk that matters in this case, not where the desk is located.