What is the way to repent and atone for paying a bribe, knowing that I am deeply regretful?

If you are predominantly certain that injustice will befall you unless you pay a bribe, then there is no sin upon you for paying it. If your predominant certainty later proves to be mistaken, we ask Allah to overlook your error and forgive your mistake, for He Almighty says: "Our Lord, do not take us to task if we forget or make a mistake," and because the Prophet, peace and blessings be upon him, said: "Indeed, Allah has overlooked for my Ummah errors, forgetfulness, and what they are coerced into." And, assuming you did fall into sin, repentance is sufficient for you, and you are not required to give charity.