How to motivate students to complete low-point homework?

Some students had determined that each assignment had such a low value, they did not bother to complete the work. These students all did very poorly on the final.

One important way to combat this attitude is to show the actual statistics from the course you just completed.

"Here is the distribution of homework completion across last semester's course. And here is the distribution of final course grades. Notice that every student who completed less than XXX% of the homework failed the course."

Repeat this exercise after every midterm, showing similar statistics for the current class. "See, here is the distribution of homework completion so far, and here is the distribution of midterm grades. Notice that every student who completed less than XXX% of the homework failed the midterm, which is completely consistent with last semester."

In other words: "I am not bluffing."

If you want to be even more direct, you can announce in your syllabus that anyone not completing XXX% of the homework automatically fails the course, regardless of their performance on the final exam. Be sure to read this sentence out loud in class on the first day.

Of course, you also need some carrots to go with the stick. The homework should not only be useful, but also interesting. The students should have access to any resources they need to master the homework material, including rapid and useful feedback. The homework should be realistically tuned to the skill and maturity levels of the students, and you should carefully discuss your expectations with instructors of any prerequisite courses.

My supervisor will not be happy to find that I am failing 25% of the students.

I assume you mentioned the abysmal homework completion rate to your supervisor as soon as you noticed it, which was relatively early in the semester. Right? So how did your supervisor respond?