Is it permissible to continue with an extra job that requires caring for a patient every Thursday and Friday, despite being unable to change these days and the patient's wife refusing to leave him alone, knowing that the patient requires continuous medical care?

If you need to work on Friday and would be harmed by not working, then this is an excuse for missing the Friday prayer (Jumu'ah), even if someone else is available to care for the patient. However, if you would not be harmed by not working, then it is not permissible for you to miss Jumu'ah because of it, unless no one else is available to attend to the patient.