My son was in county jail for about 23 days waiting to be transferred to state prison, however, CDCR doesn't have any record of that time and is not including it in calculating his release date? How can I get it sent to CDCR?

Check the Sentence Abstract, located in the court file. If it is OK, then a certified copy can be provided to CDCR. (Certified means it has the official court seal to authenticate the order.)
If the abstract is wrong, it needs to be corrected by order in the sentencing court. The trial lawyer may need to get a court date. The lawyer will need the dates of incarceration that are missing from the order.
It is possible that the credits don't go to this particular sentence. This might be the case if he had multiple cases happening at the same time. To find out if he had multiple cases you can look him up at the court clerk's office and pull all his files that were open at the time of sentencing.

My son was in county jail for about 23 days waiting to be transferred to state prison, however, CDCR doesn't have any record of that time and is not including it in calculating his release date? How can I get it sent to CDCR?

The abstract was prepared and signed before he was transferred, so it did not include the time he spent in County Jail (3 weeks) waiting for them to transport him to State Prison, so the abstract shows only in custody credits of 5 days, attributable to the charge he pled to, but nothing about the time he spent in jail after sentencing..do they usually revise the abstract after they are transported? I'm assuming that 3 weeks waiting wasn't just dead time..

He gets those credits from sentencing until arrival at reception.
The local court doesn't order any credits that happen after sentencing, so the time waiting for transport is CDCR's responsibility. If CDCR has the sentence abstract (and they almost certainly do because it was delivered at the same time as they recieved the inmate) they'll easily see the number of days credit.
If simply asking doesn't fix the problem (and it likely would), he can ask for a credit review (pointing out the missing dates when he was in custody awaiting CDCR transport.
One other possibility is to get the sentencing court to issue an order acknowledging tht he was in the local jail for a certain number of days after sentencing, and have the clerk send it to CDCR. That'll provoke CDCR into looking at the issue. The Judge isn't going to want to do that until it's clear that there's a problem that CDCR isn't going to fix on their own, and the lawyer would have to provide a declaration about efforts made to attempt correction.

The abstract was prepared and signed before he was transferred, so it did not include the time he spent in County Jail (3 weeks) waiting for them to transport him to State Prison, so the abstract shows only in custody credits of 5 days, attributable to the charge he pled to, but nothing about the time he spent in jail after sentencing..do they usually revise the abstract after they are transported? I'm assuming that 3 weeks waiting wasn't just dead time..