Is taking money from patients without fully recording it in the official ledgers considered unlawful (haram), and what should be done to clear one's conscience before God after leaving the job?

Your work in the medical center for a known monthly wage is a form of ijara (leasing/hiring), and an employee is only entitled to what was agreed upon in the contract, for Allah Almighty says: "O you who have believed, fulfill [your] contracts," and according to the hadith: "Muslims are bound by their conditions." Therefore, you must return the money you took from the center in excess of your salary. If you do not know the exact amount, act according to your best judgment, and you must repent to Allah.