Is a definite article needed before "sleep"?

Solution 1:

I actually can't think of any example where you'd use the definite article for the meaning of sleep in your sample sentence.

However, it's interesting to note that the word used for said yellowish substance is also sleep. And for this meaning of sleep, the definite article is appropriate. E.g. I wiped the sleep from my eyes.

Solution 2:

As per the other answers, the article is obligatory absent here. But you could avoid the issue by saying, instead, when you sleep. An idiomatic version of the question would be:

What's the English for that yellowish stuff that gathers in the corner of your eye when you sleep?

(How do they call sounds foreign, and the passive is gathered wrongly implies an agent who gathers, which the middle gathers avoids. Human eye and substance are both fine, but rather formal sounding.)

(BTW: the answer in my dialect of English is sleepy dust, but I know that other dialects have different names for it.)