Is one absolved of responsibility by donating public money, unlawfully acquired, to a needy relative for surgery, in cases where it is impossible to return it to the governmental entity?

As long as you have striven to return the money to the officials, and they apologized to you, citing the closure of the fiscal year, then it is sufficient for you to spend it on the poor, the needy, and the like. And if your relative is among them, then you have fulfilled your obligation.