What is the ruling on interceding for a qualified person if the intercessor has an illicit relationship with the official? Is it permissible to perform Istikhara concerning this?

Intercession (shafa'ah) is permissible and encouraged if it is for good. Texts have been revealed that urge it, such as Allah's saying: "Whoever intercedes with a good intercession will have a share from it," and the Prophet's (peace be upon him) saying: "Intercede, and you will be rewarded." The principle of intercession is the exertion of influence or power to help a person obtain their right. So, if you are qualified for the job, there is no harm in seeking someone to intercede for you. If the female worker is committed and pure, there is no harm in seeking her intercession; otherwise, no. As for Istikhara (seeking guidance from Allah), it is prescribed before every permissible act.