Well, it varies in different fanfics. The 11th Doctor could be hurt by powerful villains, or maybe in trying to save someone. It could also be from internal conflicts or emotional stress.
It could be due to various imaginative plot devices created by the fanfiction author. Maybe in a battle with a powerful villain or during a risky adventure.
The theme of self - discovery while being hurt is also prevalent. The doctor might learn something new about himself, like his limits or his hidden strengths, when facing injury or pain. It can be a journey of growth in the midst of a difficult situation, which is quite fascinating in these types of fanfictions.
It could vary depending on the creativity of the fanfiction author. Sometimes it might be in a battle with a powerful villain, or due to a risky experiment gone wrong.
In this fanfiction, perhaps the doctor monster has special powers that normal doctors don't. The hurt child might be injured physically or emotionally. If it's physical, the doctor monster could use its unique abilities to treat the wounds. Emotionally, it could offer comfort in a very different way than a typical person would.
One common theme is the struggle of the doctor while dealing with their own injury or illness while still trying to save patients. It shows their dedication.