How did they live to adulthood, not knowing how to glue a scratch with a plaster, they dragged them to the doctor for this! Or were all the characters four years old in the original script? In principle, this would explain such a stupid conflict.
Yes, it is an extremely small wound and, no, she did not need to go to the hospital because she was sneezing while wearing wet clothes. The greater point is that the male lead is concerned about her and her health. The more he is able to fuss over her condition, the more time he is able to spend with her. Look outside the box.
All of these C-dramas and K-dramas are like that, not just this one LOL. If someone trips, they end up in a hospital bed for a week. If they sneeze, they are in the ER for two days. It's kind of funny.