Giving money to an employee working in the department is not permissible, and it comes under the heading of bribery and haraam gifts to workers. Bribery is a kind of ruining the land and corrupting the people, and it is a major sin because of the report narrated by Ahmad (6791) and Abu Dawood (3580) from ‘Abd-Allah ibn ‘Amr (may Allah be pleased with him) who said: The Messenger of Allah (blessings and peace of Allah be upon him) cursed the one who gives a bribe and the one who takes it. Classed as saheeh by al-Albaani in Irwa’ al-Ghaleel, 2621.
So long as this is the time it takes to process these papers, and the delay does not result from the employee taking too long and being unfair, there is no justification for paying money. You have to be patient or discuss it the official in charge in the hope that he can expedite the processing of your papers.
No one has the right to do what Allah has forbidden on the basis that he will repent or offer expiation, because he does not know whether Allah will give him enough time to repent or accept his repentance. There is no expiation for bribery according to sharee‘ah.
And Allah knows best.