Thank you for your explanations. Your situation sounds indeed complicated and I still haven't fully understood what kind of papers the jobcenter is asking for. However, I am afraid that the jobcenter will not show consideration for your child's father's issue caused by his double life. For the jobcenter it is always important to have proofs of everything related to the financial situation - verbal explanations will not be sufficient. Therefore, you will most probably not be able to help him in this.
In any case, please don't hesitate to get back to us with any further questions.
All the best,