Dear @Zingua,
Welcome to the Wefugees Community and thank you for reaching out to us.
Unfortunataly, I never came across a case in which the acknowledgement of the paternity was made while the father was not present in Germany. Usually, both parents need to be at the local authority (usually the so-called "Judendamt") for the respective appointment and provide the required documents (see also this thread for information in English).
I tried to make some research, if the German Embassy in Ghana, for instance, is offering this service as well. However, they write on their website that they can only support in this when the father is German citizen and the child is living with the mother in Ghana. I will also try to do some further research. But for the meantime you may want to contact the German embassy directly to get the most reliable information. At the same time, your child's mother could contact the local "Jugendamt" or registry office (Standesamt) for an advice as well. As a "Plan B", you may consider to do the process when you are visiting Germany the next time.
I will also link our dear @Alla_fka here: Are you familiar with these kind of constellations and/or do you have any information?
In any case, please also don't hesitate to get back to us with any further questions.
All the best,
Meike