If you are representing yourselves, the judge might possibly read the letters but most likely not. If you have attorneys, the judge will definitely not even read the letters. The proper method to address a court is through motions and pleadings filed with a court and testimony at a hearing.
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Ms. Morcraft is correct - and the place to tell "your side of the story" is in the Motion for a Modification. Is the letter she wrote her idea of a Response to your Motion? The best thing to do (outside of hiring an attorney to protect your interests) is to wait till the hearing and advise the judge that you know that the Mother wrote a letter and you'd like to respond to those accusations. The judge will want to know the entire situation but typically he doesn't have time to sit around reading the letters that pro se litigants send him.
This response is for general purposes only and does not establish an attorney-client relationship. You should contact an attorney to fully discuss your issues.
The Judge will not be reading anyone's letters unless they are interpreted to be petitions, answers, motions or some sort of other pleading that has also been served on the other party. You have already filed your petition and hopefully had it served on the mother and you will have a court hearing to present your case. The Judge will not read your letter if it has not been also served on the other party. Present your case in Court.