Is it permissible for a Muslim young man to work on a project offered to him by his cousin, who is a manager in the same department, in order to improve his financial income, knowing that he partnered with a contractor to carry out this project?

What you have asked about has two scenarios: 1. If your cousin will prioritize you over someone more deserving or will show you favoritism that harms his administration, then he is not permitted to do so, nor are you permitted to accept it; because that would be cooperation in sin. 2. If nothing of the sort occurs, then there is no blame on him for hiring you, nor on you for accepting; because a cousin is among the relatives with whom an agent is permitted to contract.

And if the work is permissible, there is no harm in you involving another contractor with you, provided that the contract between you is free from Sharia-prohibited elements such as gharar (excessive uncertainty) or jahala (ignorance).