We understand this might be common practice where you work or in your healthcare system. As OET is a language test, one of its criteria is ‘genre & style’ which assesses your ability to write appropriately about the situation.
In English, it is polite to refer to patients by their name rather than as ‘the patient’ or ‘the client’.
You could think about it in another way too.
The person you are writing to is responsible for hundreds of patients. It is clearer for them if you use the specific patient’s name about whom you are writing in your letter. This helps them identify which of their patients you are writing about and understand the contents of your letter more quickly.
The one exception to this might be when writing to a reader who already knows the patient. In this context, it is useful to clarify the existing relationship to the reader e.g.
I am writing to update you about Mr X, your patient, who has recently…