Is the money received from working as a supervisor in an internet cafe where songs and TV series are played and newspapers are read, with the manager's knowledge, lawful?

Your work in maintaining devices, opening the hall, and closing it is permissible. The fact that some of the hall's patrons might commit Sharia (definition) violations does not burden you with sin, because you are not responsible for what is played on those devices, nor do you have any authority over it. Your work there does not involve direct assistance to someone who might commit a Sharia violation. However, if you are able to advise those who commit such violations, then you are obligated to do so as part of enjoining good and forbidding evil.