Definite Article – ‘With Eyes Wide Open’ vs ‘With the Eyes Wide Open’

Since you are talking of a specific knowledge, the article is correct. If you were talking of knowledge in abstract, then you would not use the article, as in "a thirst for knowledge" or "the transmission of knowledge."

The important part here is the difference between the indefinite article and the definitive article.. Think about the difference between these sentences.

There's a car over there.

Here's the car.

In this example, a refers to something without a specific significance, or possibly one of many. Consider finding the car, that's the one you're looking for, that's the one you were talking about or the one you're going to ride in. It's specific, and significant.

In your examples, a need refers to one of many needs that may or may not arise. The need is not specific and it has not been defined in discussion. "If there is a need" means anything could happen that causes the need, and the need may or may not exist. The exact need or factors that cause the need are not known or defined.

With the need, we're being more specific. In I see the need to buy more coffee. the need is clear: it's buying more coffee. The cause of the need is also fairly clear (though implied), because the coffee is probably gone or almost gone.