Is it permissible to pay a sum of money to an official in exchange for obtaining social housing, knowing that those seeking housing suffer from favoritism, have limited income, and do not own a home?

If you are eligible for housing and meet the conditions, and you were not granted housing due to the injustice of the officials, then there is no blame upon you for paying them a bribe to obtain your right. The sin is upon them, not upon you, provided that no harm results from this to anyone, such as taking their right or advancing ahead of their turn.