Is it likely a judge will sentence the guilty party to jail time even if they admit their wrongs and show willingness to get everything on the right track with the other party?
it is neither, it is contempt of court which has a maximum of 6 months jail. It is unlikely for a first contempt to go to jail (though occasionally it does happen). Usually the party gets yelled at and given purge conditions (things they must do not to go to jail). If they do the conditions, no jail.
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Contempt of court is a lever to force parties to do what the court wants them to do but most judges and court commissioners jail people only sparingly because the goal is to get the case resolved and locking someone up often creates more problems than it solves. (For example, locking up a parent who needs to pay maintenance or child support often leads to further problems.) Parents are playing with fire if they repeatedly thumb their nose at the court, though. Who wouldn't love to have the power to lock someone up who was repeatedly ticking you off?
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Your question is a bit lacking in facts. There are two different kinds of contempt.
The first kind of contempt is criminal contempt. For example, if you rush up to the judge and punch him in the nose. That will be a criminal contempt. The court will sentence you to a fixed jail term. If the jail term is likely to be longer than a local maximum (usually six months), another judge will hear the case and issue the sentence.
The second time of contempt is a civil contempt. This is an inherent power the court has which is used when you willfully refuse to follow a court order. The court does not sentence you to a fixed time in jail because the court believes you "hold the keys" to your jail cell. You can comply and be released.
Don't go to court without complying with the court order. Do your best to comply before you get in front of the judge.
I am admitted to practice in Connecticut and limit my responses to CT law.