Is it permissible to pay a sum of money to the judge to rule in favor of the financial right resulting from work performed without a signed agreement, and the judge rejected the case for this reason?

Bribery to a judge is forbidden for both the giver and the receiver. It is considered unlawful gain (suht) and includes bribery in judgment. If it is for the judge to rule justly, the judge becomes a fasiq (transgressor) by accepting it, and his ruling is not enforced. The briber deserves condemnation. If it is for the judge to rule unjustly, the judge becomes a fasiq from two perspectives: his acceptance of the bribe and his ruling with injustice, and similarly for the briber.

As for bribery to repel injustice, it is forbidden for the receiver (the unjust authority), but not prohibited for the giver (the oppressed). If the oppressed person cannot reclaim their right except by paying a bribe, and they have a significant need for this right such that they would suffer harm that Sharia came to remove if it were withheld, then they may be permitted to pay the bribe to attain their right. However, if their need is minor and they would not suffer significant harm from its absence, then paying it is not permissible.