Dear @Lydia,
Welcome back to the Wefugees Community and thank your for your question.
I remember that we discussed your application and the residence permit requirements for "Unterhaltsvorschuss" a few months ago here.
In general, single parents/children can also receive "Unterhaltsvorschuss" if one parent is abraod (and not paying the child allowance) or even unknown. As my colleague explained in your earlier opened thread on this topic, as an applicant you are obliged to give all information to the "Jugendamt" (which you did, apparently).
If you receive jobcenter money, it is also in the jobcenter's interest that you apply for "Unterhaltsvorschuss" which would be considered /deducted from the jobcenter's allowance. Once your child will turn 12 years old and you still receive jobcenter money, you will only be entitled to the "Unterhaltsvorschuss" if you are working as well with an income of minimum 600 EUR per month.
If they are working on the application since March it is indeed very long. But I am afraid that there is not much you can do to fasten the process as long as you provided all necessary information. Important is that the jobcenter is not making any trouble and keeps paying your full unemployment money. I recommend to keep them updated on the process. But there are no problems so far, right?
Do you agree with my explanations dear @mbeon-Éanna?
All the best,
Meike