Is it Islamically permissible to take back one's wife after she has been divorced with a single, revocable pronouncement, and what is the ruling concerning the divorce if she is taken back, given that she has repented and wishes to rebuild her home, especially after a young man claimed to have had an illicit relationship with her?

A divorce pronounced with explicit wording does not require intention. So, your statement to your wife, "You are divorced," effects the divorce, and your claim that you did not intend divorce is not accepted, especially if you uttered it after a period where you were able to speak. If you said, "You are divorced," and a long time passed, and then you repeated the statement, then two divorces are binding upon you, and your claim that you intended confirmation is not accepted. However, if the second utterance immediately followed the first, it is not counted as a second divorce if you intended it for confirmation. If this divorce is the third one, then your wife becomes irrevocably divorced (baynunah kubra) and is not lawful for you until she marries another man, he consummates the marriage with her, and then he divorces her. If the divorces are less than three, you may take her back as long as she is within her waiting period ('iddah); otherwise, she cannot return except with a new marriage contract. If suspicion or persistence arises from your wife, divorcing her is preferable; otherwise, keeping her is more appropriate. There is no need to verify the paternity of a child, as the general rule is that whoever is born on your marital bed is attributed to you, unless you deny him through 'li'an' (oath of condemnation). The husband must fulfill his duty of qawamah (guardianship/maintenance), close the doors to fitna (temptation), establish the limits of Allah, and cooperate with his wife in obeying Allah, while observing the Shari'ah's limits and manners in dealing with non-mahram women. It is impermissible to accuse a Muslim without clear evidence, and delving into people's honor is a serious matter.