Are the expenses paid by a private entity to a government employee in exchange for leniency regarding its shortcomings considered unlawful, given that the higher authority is aware of this and overlooks it?

What you take from the contractor is a bribe, and it is impermissible to take it. This is because a bribe is what is paid to a person to invalidate a right or uphold a falsehood. Perhaps the contractor, by paying this amount, aims for you to overlook his shortcomings and be lenient with him in the reports. The responsible authorities must meet the needs of their employees to prevent their exploitation, and employees must be content with what Allah has allotted to them. The Prophet, peace and blessings be upon him, said: "Allah curses the one who offers a bribe and the one who takes it."