How can prevent WhatsApp from from putting received photos in my phone's gallery?

First of all, sending patient data via WhatsApp is risky. See Is sending patient data via WhatsApp safe?.

By default, all photos you take within WhatsApp appear in your camera roll. You have to remove them manually (see previously listed instructions). You also have to remove them from the conversation.

1. Go to the chat that includes the image you want to remove.
2. Tap on the image.
3. Tap the Trash icon.

If you have lots of offending images in conversations, here is how you can clear conversations.

1. Go to the Settings tab,
2. Tap on Data and Storage Usage.
3. Tap on the chat/person you want to clear.
4. Tap on the Manage... button at the bottom.
5. Select Photos (and Videos).
6. Tap the Clear button.

Even more problematic, by default, WhatsApp automatically saves all photos you receive into your phone gallery! On iOS it is easy to change the setting to prevent this. On Android it is ridiculously hard - you have to download another app to do it!