What is the ruling of Islamic law concerning a husband who mistreats his wife after taking her back, abandons her under the pretext of pleasing his other wife, lies to her, and leaves her in a state of suspension? And what should the wife do in such a situation?

If the husband engages in such abusive intercourse with his wife, as mentioned in the question, he is abusive and unjust, and he must fulfill her rights and be fair between his two wives. It is not permissible for a husband to abandon intercourse with his wife in order to reserve his pleasure for his other wife. Patience and abundant supplication are advised, as is the intervention of wise family members to reconcile between them, so that they may offer him advice either to retain her with kindness or release her with goodness. This is because it is not permissible for him to leave the wife suspended.