The filing of a charge is a public record. Some background checks only look for convictions. Others look for only felonies or misdemeanors. Yours will probably not be found except by a very thorough search. If it is thorough they will also find that it was dismissed with no plea.
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This should not show up on your record once it is dismissed, so there should not be a need to get it expunged.
However, sometimes things do still come up on your criminal record even when they're not supposed to. If that happens, you can cross that bridge when you get there.
You can request your 'record' from the Department of Justice.
When you say 'record' - there is your CII in California. Then there are the public court records. The CII info is found via your request to the DOJ. Your arrest will likely show up. The disposition may show 'diversion.'. You will only know for sure if you request your records. I see background check company reps at court daily who pull files of folks applying for jobs. So pull your own court file to see what it shows.