No, you cannot apply for residency.
Maybe you don't consider yourself a Hondurean citizen, but it seems that you are a Hondurean citizen illegally residing in the United States.
Based on the information you provided in your post, I don't see a legal way for you to obtain a Green Card (residency).
Having a USA citizen 2-year old child doesn't put you in the positiion where you can apply for residency. A US citizen child has to be over 21 years of age in order to petition for you.
If you are married to a US citizen, he/she can petition for you. However, if no immigrant visa petition was ever filed on your behalf in the past, most likely that you will have to go back to your home country to apply for a visa.
If you’d like to schedule a telephone legal consultation, please let me know: Attorney [@] law-visa-usa.com
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