Most of the employees prefer to eat their lunch/lunches at their desk or desks.
My question is: is it lunch/lunches and desk or desks? Is the word "their" here referring to individual or employees as a group?
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This is a notoriously thorny area. But basically, lunch is preferred in your sentence. For desk / desks, it is less clear, but I prefer desks:
If you try to do a strict grammatical analysis of this sentence, you will run into problems. You have to accept that they/their is in the middle of a prolonged grammatical shift whereby it is coming to take over more singular functions.
There are other factors involved in my choices above: For example, lunch is usually an uncountable noun, whereas desk is countable. To make the whole thing more natural: