according to the Citizenship law a break in the period with a legal status (residence permit) ist not harmful if it happened because one didn't apply for an issuing or prolonging of a residence permit on time. In your case you lost a residence permit for a Duldung, so I am afraid that it breaks the necessary period of time for the citizenship. The Ausbildungsduldung is also a Duldung. But I am not sure, if there are no exclusions.
The problem with Job Center is perhaps not so bad, if you only need little additional money because of the children. In this case they sometimes make an exclusion, but it is in their discretionary power.
I would anyway consult a specialist, perhaps a good lawyer who specialises in migration law.