Although there are other answers here which get to this point, I would look at making required things required. If you need the students to do it every week, then make them do it by tying their grade to submitting it every week.
The problem is that you don't have time to assess their work every week. I think everyone here can appreciate that. However, you need to at least partially solve that issue. One key for that is the proper use of technology. Either a Virtual Learning Environment, email, or other online homework systems.
If you cannot assess all of the students, then a random assessment might get what you need. After all, as a teacher, you should be providing feedback to your students. So, have them submit their work electronically (yes, this requires a change from a paper workbook approach) and each week randomly select a few for assessment. If you see some are stars, skip their assessments for several weeks. If you see some are struggling then assess them more often. The students don't know who will be assessed but their grade might not be based on the assessment of the homework, just that they attempted it for that, you can simply scan a list of submissions. Again, you could do this with email if you have no other solution.
Ideally, the students would be intrinsically motivated and you would not have to do any prodding but in my experience some students do need a little prodding to get them to overcome the inertia of laziness. For that, submission might be enough. However, in the end, you will need to assess some. You should assess as much as you can but you have to balance this with your other needs. There's a lot more to teaching that assessment but assessment is a pretty important aspect of what we do.