There is no right to speak to the child unless the custodian agrees to it. If you do, you can dictate terms of the interview, which can include stopping the interview at any times, CPS not asking questions of the child at all, or you sitting on when they ask questions, and/or objecting to any questions and instructing your child not to answer, or simply terminating the interview right now. It is completely up to you.
Understand that CPS has no "power" per se. They cannot force an interview, and they certainly cannot simply take your child away from you without a a court order. However, they have a mandate from the state to stand in front of the Court for any child that they feel is being abused/neglected.
This means that if they feel that the child is neglected/in danger, CPS may decide to take the matter to court and ask to have the court agree that your custody rights should be suspended (or even terminated in serious scenarios) until you show to the court that you are an able custodian.
Now, while DHS has the "ear of the court," this does not mean that the court always agrees with DHS and if you challenge their claim, it may be denied. But this is a risk.
If CPS sees an issue, they will normally request that you do some custody classes and/or actions , and if not, then they can approach the court and the court may agree that if you do not, you'd risk your custodial rights being suspended or terminated.
If you feel that the requirement by CPS is unreasonable, then you can refuse. If they then go to Court (as explained above), then you'd have to show that your decision is in line with "best interests of the child."
CPS often uses misleading, cajoling, and threatening tactics, so be careful. An attorney is recommended if one decides to meet with them, because the attorney can quickly cut their attempts to intimidate the custodian down.
I am afraid you cannot demand that criminal charges be filed - this is up to the authorities. However, she can be sued for false accusation via civil court.
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