Hi @Derek
I’m neither a lawyer not an expert in family matters but wish to give you my assessment of the situation. It is not possible to simply ‘cancel‘ the shared custody rights of another parent. If one parent wishes to have sole custody, then they have to file an application to do so with their local family court. The court will **** in great detail at the situation and decide based on the best interest of the child. Both parents are involved in this process. The fact that you currently are not able to provide enough financial support is not enough by itself to remove your custody rights.
If it was the case that you had not been in contact with your child for a long period of time, had made no attempts to contact them and had made no efforts to financially contribute anything, then possibly at that point a court would assign the mother sole custody.
This article (in English via online translation) gives a good overview of the process of applying for sole custody and in what type of cases it can be given by the court:
https://***-kanzlei--hasselbach-de.translate.goog/blog/10-gruende-um-das-alleinige-sorgerecht-zu-beantragen/?_x_tr_sl=de&_x_tr_tl=en&_x_tr_hl=en&_x_tr_pto=wapp
If such a process is already happing, then I recommend you contact the organisation ZANK. They assist parents where there are issue regarding their children and there is an international aspect to the situation. They would be better able to assist you in asserting your rights:
https://zank.de/en/
Once you are back in Germany and if you have any difficulty in asserting your custody or visitation rights, you can get help from the Jugendamt or an independent family advisory service (Familienberatungsstelle).
I do not understand why you had to leave Germany. You say the birth certificate was not available. But if there was at least a recognition of paternity (Vaterschaftsanerkennung) and declaration of shared custody, then this is generally enough to be issued a permit. Here, however, the nationality and residence status of the child plays a role.
Best,
Éanna