Thank you for sharing your questions with the Wefugee Community and thank you also for your kind words!
Unfortunately, it is quite often the case, that children do not get a birth certificate but an "extract from the birth registry" if for the authorities the identities of the parents are not clear (lack of proof/documentation). Please check this thread as you will find a brochure here, which explains the registration process of a newborn in Germany really well: https://www.wefugees.de/804785/my-gambian-birth-certificate?show=804785#q804785.
One important question: Did you make a "Vaterschaftsanerkennung" for the child? This would be essential for the German citizenship. Please let us know, if you need more information on this.
Once the child has the German citizenship, you can apply for a residence permit according to §28 AufenthG as a mother of a German child.
As I mentioned already, please get back to us with any further questions or if clarification is needed. We are happy to support!
All the best,