Context #1
Andrew: Last week’s assignment was due two days ago. Why haven’t you finished it yet?
Mia: Andrew, you know I’ve been having some personal problems. Please cut me some slack.
Andrew: There are no exceptions. If I cut you some slack, then everyone will want me to cut some slack for them.
Mia: You’re so inflexible. I wish you would be more understanding.
Andrew: I would cut you some slack if you didn’t have a habit of turning assignments in late.
Context #2
Jeff: I’ve reported a student for using his phone during class time.
Michelle: Why can’t you cut the student some slack? Maybe the student had a legitimate emergency.
Jeff: He was checking facebook. How would the student benefit from me cutting him some slack?
Michelle: I guess there’s no excuse. You never cut anyone some slack sometimes though?
Jeff: Students get more than enough warnings before being reported. This student doesn’t deserve any slack. He does this daily.
Meaning: The expression "cut somebody some slack" means to give someone permission to do something not normally allowed or to be less strict about something (e.g. a policy or rule). In this example, Mia was upset that Andrew didn’t allow for additional time to finish her assignment. However, Andrew is unwilling to be more flexible because Mia has had a habit of turning things in late. So he is less willing to cut Mia some slack. In the second example, the teacher was upset that the student was focusing on his phone everyday and had been warned about the rules previously. The teacher didn’t think giving the student any more chances (cutting any slack) was of benefit to him or the student.