A world renown Cambridge Professor, takes on her new job as at a prestigious high school with a bad reputation, while she struggles as a recently single mother of two in Los Angeles.

Your logline is starting to come around. This is a much better attempt.

One thing that struck me was "Prestigious high school" vs "a bad reputation" it seems as those two things are opposite.

If both those elements are important to your story, I would keep it but if not you might just want to say "a high school with a bad reputation" and get rid of the prestigious part.

I like how this is shaping up.