The word itself will vary depending on who is delivering the service.
Many professionals who offer services will refer to them as cases when talking about the overall situation. This includes medical professionals, legal professionals, social workers, and others.
Public defenders are routinely overworked and can have an overwhelming case load.
Health care workers (doctors, nurses, physical therapists, psychologists, acupuncturists, etc.) have patients. Some mental health professionals may refer to their patients instead as clients, to avoid the implication that there is something medically wrong with them.
The doctor-patient confidentiality protects communications between a patient and his or her doctor from being used against the patient in court.
The renowned psychiatrist primarily was a professor at the medical school, but still occasionally saw clients on the side.
Social workers who work with adults have clients. Certain social workers work with families. Others work with children. School counselors work with students. Still others work with veterans. With each, the more specific term is often used in place of client.
When working with clients, social workers must maintain clear boundaries to assure professional integrity and responsibility.
Today's school counselors help all students in the areas of academic achievement, career, and social/emotional development.
Parole/probation officers have clients, but these are also frequently called offenders.
While GPS units can help monitor offenders, probation and parole officers must use them in conjunction with other supervision methods.
Police officers serve the public, but individual situations that require a police response are calls. Individuals who are arrested by the police are referred to variously as perpetrators, suspects, or defendants, depending on the context, and whether they have been charged with a crime.
Two Falmouth police officers were wounded responding to a call Friday afternoon, and a suspect was shot multiple times by the officers.
In your specific example, you can use client or case depending on the context.
My client has severe anxiety and has trouble talking with anyone he doesn't know. He's a difficult case because sometimes he refuses to even answer the door when I come by.