Avoiding the use of "the reader"?

Something you can do to trim the number of uses is to simply dictate what the author was saying or doing:

Foucault said [x].

This means [x].

This passage is read as [x].

The power of this passage reflects [x].

In other words, instead of saying, "Foucault was expressing power to the reader" just say, "Foucault expressed power."

"The reader" is typically implied by context and can often sound more passive than simply declaring what is happening. Removing "the reader" refocuses your statement on the author, the author's intent, and the writings or passages themselves.

That being said — and as others have pointed out — there is nothing inherently wrong with referring to the readers. It is good to trim a few of them out if you are feeling that it has been overused. What I have suggested is simply one way to do that.

You also have a great advantage in the sense that you have a teacher who can simply tell you what to do. Part of their job is to explicitly tell you how to write these essays and any personal preference they have should be immediately reflected in any essay you turn in to them. Simply ask after class or during office hours. If the class has a TA, they are also a good candidate for input.

It sounds as if you are also going to end up using the passive voice ("how a reader is being written to or directed in the writings"). Recasting the sentences using active voice might help you find the right object for each verb.