A mechanism and procedural rule exist by which you can attempt to do so, but if literally everything went perfectly for you, exactly as planned; and each domino fell as you wished in sequence, the end result is that you would be returned to your previous positon, prior to the plea, pending trial, but having already served a sentence. The sentence you assert by the way, makes no sense, as battery is a misdemeanor, and you could not have been sentenced to state prison on a misdemeanorbattery charge.
What was it in particular that made it obvious to you that your P.D. was incompident? Your claim for relief, by the way, would be a motion to vacate plea and resulting judgment and sentence because you assert that you were improperly advised in one or more respects or induced improperly to accept the plea, all relating back to the ineffective asssistance of your counsel, as you assert.
You are going to have a very tough time with this one, although anything is possible, and you have provided no factual basis for your claim that your counsel was ineffective, making your position almost impossible to evaluate. It is a reasonable assumption that the court colloquoyed you on the rights which you were waiving in entering into the agreement, and assuming that the court conducted itself properly, which seems like a fair assumption, you were in all likelihood specifically asked whether you were satisfied with the representation which your counsel had provided to you. If you said "no" the court would have declined to accept the plea. Therefore you are going to have to prove that you could not have learned of the basis for your claim until after the fact, and that it would have been unreasonably difficult or impossible for you to have learned of the deficiencies before entering into the plea agreement.
Again, what were some of the specific issues of concern to you?
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Unlikely. You have a very short amount of time to vacate a sentence. Withdrawing a plea, post sentence, requires a demonstration that allowing the plea to stand is a manifest injustice. Therefore, withdrawing a plea after sentencing is reserved for only the rarest of situations.
This is not to be considered legal advice nor does an attorney-client relationship exist.
You can withdraw your plea if you are able to assert that a manifest injustice was committed and/or that you were not properly advised on a material right by your attorney. However, the effect of having your plea withdrawn just allows you to start again from scratch, in which case you will have to either go to trial or renegotiate a plea.
Withdrawing a plea after sentence is over would in all probability be impossible because time limits on filing post-judgment motions would have almost certainly run out. By the way, I strongly disagree that most public defenders are incompetent.