Is a wife considered under her husband's authority after an irrevocable minor divorce, if the husband then verbally took her back during the waiting period and they had intercourse, without her family's knowledge?

If you divorced your wife with a first revocable divorce (ṭalāq rajʿī) without khulʿ, it is a revocable divorce, not an irrevocable one (ṭalāq bā’in), even if the authorized marriage official (ma’dhūn) informed you otherwise. An irrevocable divorce (ṭalāq bā’in) only occurs for a specific reason, such as khulʿ or a judge's ruling.

If you take her back (rajʿah) before the expiration of her waiting period (iddah), she returns to your marital bond. Her family does not have the right to prevent her or marry her to someone else; if they do, her marriage is invalid.

However, if you attempt to take her back after the expiration of her waiting period (iddah), the retraction is not valid, and she is free to marry whomever she wishes. You can remarry her with a new contract and dowry, with her consent, the consent of her guardian, and witnesses.

The iddah (waiting period) for a menstruating wife is three menstrual cycles, and for a non-menstruating wife, it is three months. For a pregnant wife, it is until she gives birth. It is advisable to consult the Sharia court.