You would have to have a basis to be able to "go again" and "having a court appointed attorney" wouldn't be enough of a basis. If you plead you would only be able to "go again" if you could show that your attorney was ineffective (a very hard thing to show). If you lose in trial, your conviction would only be overturned on appeal if there were things that went wrong in trial and they were wrong enough to deserve a retrial. Most court appointed attorneys are excellent, dedicated to defense...
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Under the current law, you can't vacate (or remove) a crime against a person from your record.
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