Q: "Sleep through a single night" vs. "sleep a single night" 
For the next two weeks he did not sleep through a single night.

Can we recast the sentence as follows?

For the next two weeks he did not sleep a single night.

That is, is the use of through mandatory? Is there a difference between these two sentences?

A: They do not mean the same thing.

For the next two weeks he did not sleep through a single night.

This means that he did sleep, but he did not sleep a whole night without waking up. In other words, he slept very badly.

For the next two weeks he did not sleep a single night.

This means that he did not sleep at all during a night. Possibly because he had to work, or study, or whatever. He did not sleep badly, he did not sleep at all (at least during the night - if he worked at night, he might have slept during the day).
