His Trash, The Throne's Treasure

Eleanor wants the man she is engaged to. She has loved him for five years, since she came back from the countryside to the noble house whose legitimate daughter she is, and he looks down on her for where she was raised. When the court holds its examination for women officials he sits in judgment, favors the daughter her family adopted, and fails Eleanor over a technicality. The failure sends her into the palace as a bride for the emperor, whom the court calls a tyrant. In the palace Eleanor works her way into the emperor's confidence, and the two of them go after corrupt ministers and the rigging of the imperial examinations. Her former fiancé comes round to regretting the examination some time after that.