That internal area is called a "compartment".
(If it had a container that you could slide out to place your things inside more readily, that would be a "drawer".)
(A "hole" is too generic of a term, and would not be associated with the desk; a "slot" would be a long, narrow hole that allows you to insert, for example, an envelope into an otherwise closed compartment.
This desk has a (closed) drawer with a slot.)
ell.stackexchange.com/users/87/Hellion
I would use one of the following terms: • pigeonhole , “A nook in a desk for holding papers” • recess , “An inset, hole, space or opening” • cubbyhole , “A small compartment; a pigeonhole” • desk pocket , with pocket used in the sense of a receptacle, indention, or cavity
ell.stackexchange.com/users/146/James Waldby - jwpat7
The hole in that desk can be called a slot.
See this definiton .
ell.stackexchange.com/users/1425/Tristan
One could call the "compartment" a cubby , a cubbyhole , a pigeonhole (although a pigeonhole is usually smaller), or a recess , but the simplest, most common English word might be shelf .
ell.stackexchange.com/users/43789/Torsten
Apart from anything else that does not look practical.
What would you put in there?
Paper?
Exercise books?
I would not describe that space as being a slot.
A slot is a snug, tight hole, usually horizontal, and just enough space toslide something in.
A coin for example.
A slot machine, for exampleis another name for a fruit machine.
In that case the coin slot isusually vertical.
I wouldn't call it a hole.
A hole tends to be round, not always, but if you were to ask someone to drawa hole in the wall it would be round shaped.
Personally I would call it an "open drawer" or a "paper drawer" or"an open unit" or a "paper storage nook".
Google provided thispossibility: Open paper storage drawer
The actual piece of furniture is called "a four drawer open paper storage".
But admittedly it is a bit of a mouthful.
It would take you longer to say to a friend: " Please put the book into the open paper storage drawer." than to put the actual book away yourself.
"Hole" then would be fine, life's too short.
ell.stackexchange.com/users/1694/Mari-Lou A
This question has been created to split two questions previously asked  here    It is a "hole" in a desk. What can I call this? Can I say "Put your books into the hole"?
What is the “hole” in this desk called?
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ell.stackexchange.com/users/307/Matt
A hole in a desk can be called a slot, a compartment, a cubby, a pigeonhole, a paper drawer, a nook, or a desk pocket. You could call it a slot, cubby, an open drawer, compartment, pigeonhole or nook.
You could call it a slot. You could call it a cubby, an open drawer, compartment, pigeonhole or nook.