Okay here we go again, folks. Some people on these sites sometimes wish to jump in here and tell you that you are about to be subject to the ultimate prosecution power of the law. They quote laws that describe what might happen to you IF after an investigation, the agency recommends you for prosecution. But after working many years in fraud investigation of unemployment claims and the entire unemployment insurance process, let me assure you of one thing based on my first hand experience.
In the circumstances you describe, receiving your first letter, you will not be prosecuted immediately, you do not need to retain an attorney, THIS IS NOT at this time, a criminal matter. You will be talking to no one in the D.A.'s office. This is an investigation by the fraud unit of an agency, NOT a criminal prosecution or even an investigation into a criminal matter at this time.
What you have done is remarkably common. Very STUPID but remarkably common. A lot of people who are receiving unemployment benefits think they're really needy and really need this extra money, and that they're the first people that have ever thought of doing this. Yes, sure, they told me to report any work I'd done during the weeks, but this was only part time and hey, I really really needed the money, so maybe....
They try to defraud the agency by not reporting work they did while drawing. A cross match of any wages reported under your particular social security number with unemployment benefits happens automatically to everyone who is drawing benefits.
They can tell how much you worked, when you worked it, everything that you were paid. Eventually it even cross checks all claimants with your income tax returns so if you weren't caught by a simple wage cross match, they'd eventually catch that you were paid something unreported then. So it is totally futile not to admit you did it or to try to question the accuracy of their evidence. Pretty much the system catches you, you can't get away with it, and you are caught before you turn around.
I'm surprised they didn't catch this sooner. Since unemployment benefits are NOT a needs based, poverty required program, the "gee I was so poor and I really really needed the money" is never appropriate or useful. But you do need to do what their instruction letter tells you. Don't complicate it, don't try to be smart, or lawyer up at this point or obfuscate and claim you didn't do it or refuse to admit guilt. That's just uncooperative bologney from somebody who got bad advice and it irritates the agency representatives you will be dealing with. It makes them much more inclined to push you, to threaten you with prosecution.
The very best thing you can do is contact them, and say, "I did it, and I'm sorry. I am ready willing and able to pay this money back right away." Any excuses, arguments, or legal wrangling you try to do with them at this point is superfluous. You did it, they know you did it, and believe me, they are much more interested in getting their money back than in recommending you to the D.A.'s office and requesting that you be prosecuted.
So you will receive an official overpayment decision, and you will work with the fraud unit to work out a repayment plan, with penalties. Be sure you do not 'over-promise' what you are capable of repaying. If you are financially unable to repay the money at this time, which will probably not be the case since you are working again, in some cases a waiver of overpayment, or postponement of overpayment can be granted. But if you don't need it, don't bother. You need to begin some sort of repayment plan that you have worked out with them as soon as possible. Since this was clear cut deliberate fraud there will probably be substantial monetary penalties as well as the basic overpayment.
I will repeat myself. This is not a criminal court issue. This is not even a fraud case yet, this is an investigation by an agency representative, not the criminal justice system, based on information the agency has received about wages you have been paid and have not reported. And you definitely have committed clear cut unemployment fraud. They know you did it, you know you did it.
But you are not about to be sent away for this, not at this time. Talk to them, get in touch with them immediately, and work with them. They have heard this all before, they deal with hundreds of people like you every week. I will repeat, they are much more interested in getting their money back than they are in filling the jails to overflowing with clueless and dishonest dorks who tried to beat the unemployment system. Do it again, do it on a large scale, ignore their requests for meetings, make them have to run you down, lawyer up and come in with all your defensive legal guns blazing and your chances of being recommended for an actual prosecution rise somewhat.