Dear Dr Vincent,
Welcome to the Wefugees Community and thank you for sharing your question.
Please have a look at the following thread, in which a very similar case was discussed earlier. You will find many relevant information here. For example, that you either need to wait for the outcome of the appeal or take the appeal back (which should be carefully elaborated/discussed with a lawyer), if you want to apply for a residence permit related to your children now. You can not apply for asylum and a residence permit for another reason at the same time.
You may also want to check this thread for further details on a residence permit for the sake of caring for your children.
However, also do not hesitate to get back to us if you still have questions regarding your case.
All the best,