You could ask the governor for whatever you'd like, but with your facts it is doubtul that you would get a pardon from a governor, to say nothing of qualifying for lesser forms of relief.
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You should consider getting off probation and using other statutes to clean your record that may get you to where you want to be. Look at California Penal Code section 1203.4. It may give you some answers. If you go that route you may reach your goal. A pardon is very difficult. From the few facts you set forth here it is doubtful that a pardon would be granted.
Every case is different. Specific facts of your case must be applied before relying on an answer to a general question. The answer given here is a general statement of the law on the issue that you presented in your question.
Your situation will be much easier to resolve if you hire a locally experienced defense attorney. It doesn't sound as if a pardon is the best route for you to take, but there may be other options available to you. Best of luck.
It is difficult to provide an answer on these limited facts. What firearm violation were you charged with, and have you been convicted. Depending on the specific facts you may be eligible for an expungement, but it is difficult to say if that would enable you to become a CO.