Translation:There are many books on the students' desks.
Sure. Every íróasztal is an asztal, so you're free to just call it asztal for short. Works the same in German with "Schreibtisch" and "Tisch", but not in English where you use two different base words, "desk" and "table".
But I think "tables" is accepted as an answer, and you're free to translate it with íróasztalain, too.
This whole possessive topic is not the easiest in Hungarian to explain. But to try to establish a rule for learners of Hungarian, we can probably say this: you only use the "-k" ending if there is no noun, or there is only "ő", immediately before the possessed. You can see this pattern in the above examples.
Check here for more info:
az ő asztalaik (!)
a diákok asztalai (!)
So, if the possessor is obviously plural (because -k is on the end of asztalai ), then the possessor can be expressed as a singular (ő )? Is this only for the 3rd person pronoun? And if the possessor is in the plural (diákok ), then the possessed item can be singular?
Are these both examples of plural possessors owning multiple objects, that is, "their tables" and "the students' tables"?
Not exactly like that. If the possessor is a noun, then there is no "-k" on the possession. That is, it acts like the third person singular. If the possessor is either not mentioned or it is the personal pronoun, then there is a "-k" on the plural possession.
Bottom line, check the possessor:
1- not mentioned:
2- personal pronoun "ők":
Az ő asztaluk, az ő asztalaik
3- a noun:
Péterék asztala, asztalai
If you check the comments at similar sentences, you will find several examples where this is explained in a bit more detail. And yes, it is only the third person plural that is such a special case.
As a little cheat, you can note that there can be no more than one "-k" ending in a possessive structure. Thus, the possessor and the possession will not both have a "-k" ending. The personal pronoun, for example, will lose its "-k" ending.