Is it Islamically permissible to recite the Basmala over the entire Qur’an for six consecutive recitations, leaving the seventh until a need is fulfilled, in order to hasten the fulfillment of needs, and is this permissible for someone who is not proficient in Qur’anic recitation?

There is no basis for the aforementioned image. The questioner is advised to fear Allah, engage in supplication (Du'a), remembrance (Dhikr), seeking forgiveness (Istighfar), repentance (Tawbah), and frequently reciting the Qur'an, especially during times when supplications are answered. For Allah answers the caller's supplication and removes harm from them. The Qur'an is guidance, mercy, healing, and light. A Muslim should draw closer to Allah by reciting it and worshipping Him through it, then supplicate to Allah and ask Him for their needs upon completing the Qur'an.