Praise be to Allah.
If the administration of the clinic has given this doctor permission to do scans for some patients for free, then there is no blame on you for what she did. But if that was not the case, then it is not permissible to use the clinic’s equipment without paying a fee, because the Prophet (blessings and peace of Allah be upon him) said: “It is not permissible to take a Muslim’s wealth except with his willing consent.” Narrated by Abu Ya‘la; classed as sahih by al-Albani in Sahih al-Jami‘, hadith no. 7662.
If the administration of the clinic allows the doctor to charge a reduced fee for her relatives or friends, then you should deduct the discounted amount, then pay the rest to the clinic.
You should persuade this doctor to accept the payment from you and give it to the clinic, after you explain to her that this action is haram – if the clinic did not give her permission to do that.
If she does not do that, then what you must do is make sure that this money reaches the clinic.
An-Nawawi (may Allah have mercy on him) said in Sharh al-Muhadhdhab (9/428):
If he has unlawful money and wants to repent and absolve himself – if the money has a specific owner, then he must give it to him or to his proxy, and if he has died, he must give it to his heirs. End quote.
You do not have to tell them the reason why they are entitled to this money; rather the objective is to make sure that the money reaches them by whatever means.
Please see also the answer to question no. 31234 .
And Allah knows best.