Under these very specialized circumstances, you will need to verify everything about the possibility of resuming unemployment benefits with the California unemployment system.
Usually if you accept another job, even if it is much less than what you were making at your old job, as long as it is paying more than your weekly unemployment benefit was, (gross wages, no deductions for expenses) from thenceforth, it becomes your separating employer. So if you quit this job, because you cannot afford the child care, you very likely will be considered to have voluntarily quit for a personal reason. No unemployment.
But please talk to someone in the California system. They might have some sort of special provision, or extra leniency about these assisted child care programs. As I said, most states don't have anything like this, and so I am quite unfamiliar with how these tie in with regular unemployment benefits and leaving your jobs.
I know calling them and talking to a living human being will be very difficult, if possible try to call the "technical services" or "technical assistance" offices in the department that handles unemployment, not just the regular toll free lines, as they usually don't answer, do not answer general questions.
But it looks to me very much like you wouldn't be able to draw again if you quit this job, so think long and hard before you do quit, and get the information about drawing directly from the horse's mouth, in other words, an unemployment person in CA who can look up your old claim, tell you how much you had left on this claim to draw and how their particular system would look at you quitting your job due to not being able to afford child care.
I am sure that that what California's somewhat more generous assistance programs used to be are still no gravy train, but income based child care assistance is quite a rarity throughout the nation. But unemployment insurance is not a needs based program, so if you accepted another job and are working there and making even minimum wage, you're considered to be fully employed, no longer entitled to what you made on the old job, and if you quit, it's considered to be a personal choice.