Manga where NEET is forced to go to high school to gather information about girl
It is Doubt! by Sakuya Amano.

Not to be confused with an entirely different series Doubt by Yoshiki Tonogai

Synopsis


  Ichiru Kanzaki is given an offer he can't refuse: to clear his entire debt, he has to enroll in the high school where his father works to track down his father's illegitimate daughter, his own younger half-sister, whom he has never met. But which girl is she? 


The description in the question matches the events in chapter 1 of the manga.

Below are two pages towards the end of the chapter, showing the main character's mother and her background (owner of a men's club and well-known by the police).