These expressions are some of the most difficult to master in English. Let's examine them more closely: To run behind schedule/ Running behind schedule 'Running behind schedule' is what's commonly referred to as a set phrase in English. It means to arrive later or to have something happen later than expected in the future. To rephrase some of what you wrote, I'd expect to hear: We're running behind schedule on (progressing through, finishing) our course syllabus (people, students, class), so we have to (try to) catch up. In colloquial North American English: we often use the verb 'to skip.' Professor: We're running behind schedule on the course syllabus, class. So we're going to (have to) skip some sections (in order) to catch up.

This sentence is perfectly fine: We'll have to speed things up and I will have to cut out some topics so we can catch up on our syllabus. These are also OK: We'll have to speed things up and I will have to remove some topics so we can catch up on our syllabus. We'll have to speed things up and I will have to eliminate some topics so we can catch up on our syllabus. We'll have to speed things up and I will have to skip some topics so we can catch up on our syllabus.