First, let me note that “homework” is an uncountable (mass) noun in English. It is a common mistake, even among some non-native teachers of English as a foreign language, to say things like, “Hand in your homeworks, please.” The word “homework”, being uncountable, does not have a plural form. If you really must refer to several separate “pieces” of homework, the word you are looking for is “assignment”:
It is also not possible to use “homework” with “a”:
Let’s move on to the main topic of this article. Without further ado, the correct verb to use with “homework” is “do”, not “make”:
“Homework” is not in any way special in this respect. We always use “do” with activities and “make” with objects that are being made. Just like we “do work”, not “make work”, we also “do homework”, meaning the activity.
The confusion arises from the fact that the most direct translation of “homework” into other languages often refers to the sheets of paper on which the assignments are written, hence it would make sense to “make it”. For the very same reason, we wouldn’t usually speak about “writing homework”: