I have found 2 bugs with this minigame.
First, don't save after you heal the little girl but before launching the minigame. Because you can't heal the sick people then. Clicking on them just play their dialogs.
Second, the counter is buggy and counts 2 too many. When I heal everybody, it's still count 2. I can't finish the minigame, and some of them get sick again...
Is there a way to progress the story without going to the clinic ?