Is the money I receive permissible, given that I succeeded in the employment exam after being recommended and having my scores raised by three grades, noting that I had previously refused that?

The person who failed the exam, if interceded for and given undeserved marks to pass, then this act is impermissible; because it involves cheating and deception. As for the work resulting from this success, she is permitted to continue in the job as long as she is qualified for this position.