What is the ruling on an agreement between doctors and a hospital director to be present only one day a week for each doctor, while recording daily attendance, especially since the salary paid is modest?

If the manager authorized your absence and the presence of one doctor was sufficient, then there is no objection to not attending, and the salary is lawful. It is permissible for the manager to record your daily attendance. However, if he was not authorized, or if one doctor was not sufficient, then absence or taking pay for it is not permissible, and it is not permissible for the manager to record attendance. The smallness of the salary does not justify violating the contract. The higher authorities should be informed if the number of doctors in the hospital exceeds the need.