Employment Law Questions? Ask an Employment Lawyer.
If you were working, you have an obligation to report those earning to unemployment and failing to do so could get you into trouble with the state. While it's technically possible to criminally charge that as fraud, the more common result is just a civil action.
Earnings do effect your unemployment rights, and anything you earned then would have been offset against your unemployment earnings. Quitting the part time job without good cause could also be used as a basis to disqualify you for continued unemployment, at least for the offset. They absolutely will take that into consideration. While it is unfortunate, it is better than the alternative.
If they find out without your reporting the income, you would be entirely disqualified from continued unemployment benefits both now and for a time in the future (if you were to get another full time job and then leave that under circumstances allowing unemployment) and potentially fines on top of that.
While there will be financial loss, the best policy here is reporting the income.
If they are doing as they are supposed to do, they report it immediately and every time. They report it and the state has other means or sources of determining earnings.
Unless you were paid under the table, there would be a record created which the state would become aware of.