What is the ruling regarding a wife's performance of Tarawih prayer while her husband remains with their child, given that he did not pray 'Isha in congregation due to staying with the child?

Congregational prayer in the mosque is obligatory for men, and Tarawih prayer is a confirmed Sunnah (definition) that should be prayed congregationally in the mosque. A woman's prayer in her home is better than her prayer in the mosque, and this includes Tarawih prayer and others. The husband's action is regrettable because he abandoned his obligation (Maghrib prayer in congregation) and abandoned the Sunnah (Tarawih prayer in congregation) for something permissible for you. He should not repeat this and should enable his wife to pray at home.