I have absolutely no idea what you mean about being "charged" with abandonment. Without having the papers in front of me, it is impossible to comment. I will say that this sounds very serious and you are making a mistake if you do not find some way to come up with the money to hire an attorney who can sit down with you, review everything, and give you the guidance it sounds like you need.
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Per Georgia law "'Abandoned' means left without provision for reasonable and necessary care or supervision".
Of course, you would be better off being represented in ALL of your pending proceedings. However, in the event that you cannot, you must be prepared to present a defense to the allegation that you have failed to provide for your child's care. Paying the past due amount will help. However, it will only help a little since it will have been paid after the the complaint/petition was filed. You would need to also provide proof of the other forms of care that you have provided (including the time that you have had the child in your custody). (In case it was not clear... you will have to appear for court on the Abandonment charge even if you pay the arrears prior to that date.)
The above answer is a general explanation of legal rights and procedures. It does not constitute legal advice. Nor does it establish an attorney-client relationship between the individual posting the question and the attorney providing the answer.
If I understand you correctly, you have filed for contempt against the mother and you have also been charged with abandonment, which is a criminal charge. On the abandonment charge, you will need to talk to the solicitor or DA assigned to your case to see what your options are. Also, depending on the county there could be an indigent defense (or court appointed attorney) system and you could request representation that way. Whether the court grants your request depends on the requirements of that court. At any rate, if you make any kind of payment of child support make sure that it is documented (check, etc) so that you retain proof. Best of luck to you.
My comments are not intended to establish an attorney-client relationship, are not confidential, and are not intended to constitute legal advice. Proper legal advice can only be given by an attorney who agrees to represent you, who reviews the facts of your specific case, who does not have a conflict of interest preventing the representation, and who is licensed to practice in the appropriate jurisdiction where the legal issue may be filed or in the state where the law applies.