He probably has the right to designate someone to take care of the child, but the court would likely grant a right of first refusal if you properly requested it through a modification. I would have to look at the paperwork to be sure. The right of first refusal is typically done so that if the parent will not be with the child for 2 hours or more, the other parent has the right to the child at that time before anyone else.
Disclaimer: This answer is intended for educational and informational purposes only and should not be considered legal advice nor forming the attorney client relationship.
Most single parents must work to support themselves, and this is true for both custodial and non-custodial parents. Custodial parents hire babysitters and daycare providers to take care of the children while they work, and non-custodial parents must do the same if they must work during a period of possession of the children. The more disturbing aspect of your question is your statement that a 9 year old child is occasionally being left home alone while the non-custodial parent is working. You do not mention how long a span of time the child is being left alone, but many parents and judges would agree that leaving a 9 year old at home alone for more than a relatively short period of time without appropriate safeguards is dangerous.
If your concern is that your child is with a third-party caregiver while the father is working instead of with you, there is little to be done about it. If your concern is that the child is not being properly supervised by an adult while the father is working, you should contact an attorney to discuss your options.
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