What is the ruling on specifically mentioning a person's name and their problem when supplicating for them in prayer?

Supplicating for a specific person during prayer is permissible. The Prophet, peace and blessings be upon him, supplicated for individuals by name, and also against others, as he did in his Qunut after bowing, when he supplicated for "Al-Walid ibn Al-Walid, Salamah ibn Hisham, and the weak among the believers," and supplicated against Mudar, openly declaring it. Just as supplication is permissible, so is mentioning its reason without elaborating, like his supplication on the Day of Badr: "O Allah, they are barefoot, so provide them with mounts. O Allah, they are naked, so clothe them. O Allah, they are hungry, so feed them."