Praise be to Allah.
First of all we should examine the way in which projects are awarded by the state through direct appointment. If this is done in an open and transparent manner, without going against the regulations set out by the state, then there is no problem with it. But if it is done in a manner contrary to the regulations set out by the state, then this is not Islamically acceptable and it is not permissible for the employee to embark upon it.
If it is Islamically acceptable, then we must examine the situation of the employee (who was awarded the contract):
If he was acting as an agent on behalf of the government department, then he must return the surplus money and must try his best to do that; if it is not accepted from him, then he should spend it in the public interests of the Muslims.
However, if he was a contractor appointed by the government department to carry out the project – and the regulations allow that – then the extra money is his and is regarded as part of his profits. For example, if they appointed him to construct a building for one hundred thousand, and he managed to do it for eighty thousand, then the extra money is his.
As you can see, this answer is in general terms, as was the question. If you would like a specific answer, then we need more details about the matter.
And Allah knows best.