As described here, you may be able to drive with your child in the car when you have a provisional license, if you have a letter from one of the adults described.
Here is an example of a letter that a Principal wrote for a student who needed to drive herself and her daughter to school.
To Whom It May Concern:
My name is ___________________ and I am the school principal at Garden High School. Sara Sue needs to transport her child, Alia Sue, in her car to and from Garden High School on school days for the school year of 2014-2015. Sara needs to drive Alia because there is no bus service available and no other way for Sara to get herself and Alia to school. Under California Vehicle Code §12814.6(b)(2)(B), individuals with provisional licenses can transport minors in their cars without supervision for school or a school-related activity.