Is it permissible for a person to request to lead people in prayer in the mosque if those who are leading them are not proficient in reciting the Qur’an, and what is incumbent upon him to do?

A person may seek to be an Imam if there is no one more deserving than him. Some Companions, such as Uthman ibn Abi al-As (may Allah be pleased with him), sought it. The hadith indicates the permissibility of seeking leadership in good, and this is not part of the reprehensible seeking of authority. Rather, it is like the supplication of the Servants of the Most Merciful: "And make us leaders for the righteous" (Quran 25:74). It may even become obligatory for him to assume it if no one else is fit for it. As for how to obtain it, that depends on the authority responsible for such matters in the relevant place.