Is it permissible to go to the Haram for prayer and supplication to achieve a specific matter, such as the healing of a sick person or being blessed with children, and is it permissible to fast for the same purpose?

There is no impediment to intending to go to the Sacred Mosque (Masjid al-Haram) for supplication (Du'a). However, it is preferable that the primary intention be to worship Allah and seek reward (Thawab), with supplication being secondary to prayer (Salat). The Sacred Mosque is a blessed place that combines the virtue of the place with its state. The Prophet, peace and blessings be upon him, commanded supplication during prostration (Sujud) because it is most deserving of being answered. Supplication is answered at specific times and places, including during prostration, when rain falls, after the Adhan and Iqamah, at the time of breaking the fast (Iftar), in the holy sites (Masha'ir), and in mosques, especially the three mosques (Masjid al-Haram, Masjid an-Nabawi, and Masjid al-Aqsa). Seeking intercession (Tawassul) through righteous deeds, such as prayer and ablution (Wudu), is permissible. Among the reasons for the acceptance of supplication is that the worshipper precedes it with ablution and prayer, as well as fasting. The intention behind fasting is to worship Allah and attain reward, and the supplication of a fasting person is answered, especially at the time of breaking their fast.