Welcome to the Wefugees Community and thank you for reaching out to us.
When applying for asylum at the "Bundesamt für Migration und Flüchtlinge" (BAMF) in Germany, the BAMF is assessing the reasons for the application on an individual basis. People facing persecution (based on different aspects as race, gender, ethnicity, religion etc.) in their countries of origin should be granted asylum or regognized as refugees. If facing serious harm (due to violent conflict, for instance) in case of returning, people may be granted subsidiary protection.
I found a summarizing explanation on infomigrants.net:
Refugee status and subsidiary protection are both forms of international protection granted to asylum seekers. It is not up to the asylum seeker to choose which one to apply for – the authorities decide which status they will grant. There is a European Directive (a non-binding rule) laying out standards of protection, but it is up to each country in the European Union (EU) to incorporate them into its own laws, so there are differences among member states.
As a rule, in order to be granted refugee status a person must meet the definition of a refugee under the Refugee Convention (‘Geneva Convention’) from 1951. An important aspect of this is whether refusal of the application for refugee status would result in the person's "refoulement" (illegal forced return) to a country where their life or freedom would be threatened. If so, they they will be recognized as a refugee.
Subsidiary protection – also called humanitarian protection – is a lesser form of protection which may be given when a person does not meet the criteria for refugee status. To qualify for subsidiary protection, a person must usually prove that they would face a risk of serious harm if returned to their country of origin. It is not necessary to be at risk for a specific reason such as race, religion or political opinion.
Health issues are not per se a reason to be entitled for the above mentioned types of humanitarian protection. However, they might present a reason not to be asked to leave Germany or to get deported. In case your or your child's application for humanitarian protection got rejected and there are serious reasons, why it is not possible for you to return to your country of origin, I would highly recommend you to get support by a lawyer or a migration counselling office. Please let us know if you need support in finding one in your area and don't hesitate to get back to us with any further questions.
All the best,