If you want to add an extension field to the standard c4codatapi service via Cloud Application Studio, you will find that "c4codataapi" is not listed in the ODATA dropdown when you create the ODATA Extension (.xodata) object.
The solution is provided in this help article: OData API v2 Reference. The c4codataapi service is just a wrapper and you need to add the extension field to the respective child service, e.g. registeredproduct in case of Registered Product (the link provides a mapping).