Is it permissible to secretly take back a divorced wife, knowing that there is an engagement to another girl, and is it permissible to conceal the matter of the reinstated wife from the new wife?

If the wife is righteous, you are not obligated to obey your mother in divorcing her, because obedience is only in what is good (ma'ruf), and divorcing a righteous woman is not part of it. However, since the divorce has already occurred, do not take her back except with your mother's consent. If your mother does not consent, then leave her and marry someone else. Do not marry her secretly, so that conflict does not arise again in the future. If you marry a second time, you are not obliged to inform her, though it is preferable to inform her.