What is the religious ruling on giving a wife one last chance after the husband discovers her repeated betrayal and her continuous lying to conceal her actions, given the presence of children and the persistence of doubts?

Your oath of divorce to your wife that she would not return to messaging a non-mahram is a contingent divorce; if she messages the non-mahram, the divorce occurs.

The wife claims she messaged him, interpreting that she thought the intention was the internet and not the phone. There is disagreement regarding the acceptance of this interpretation. If it is said that the divorce is binding, then you may reconcile with her if this was not the third divorce.

As for keeping your wife, if you sense sincere repentance from her and a cessation of messaging non-mahrams, then there is no harm in retaining her and forgetting what has passed. However, if you perceive deceit from her and persistence in wrongdoing, then it is better to separate from her amicably, while concealing her secret and not divulging it to anyone.

The claim of sorcery cannot be delved into, neither by denial nor affirmation, due to lack of knowledge concerning it.