Is the money taken from the Yemeni worker lawful or unlawful, knowing that it was presented as assistance for the institution's rent and supplies after his non-compliance with the partnership agreement became apparent?

The issue of the institution and its obligations is specific to you, and your relationship with your friend is one of sponsorship for which it is not permissible to take money. However, it is permissible to take money for fees or work you perform. Therefore, the amount your friend paid you to remain under his sponsorship and for work outside your institution is not halal and must be returned to him. His retraction of his promise to partner does not make this amount permissible.