Praise be to Allah, Lord of the Worlds.
There are two cases as regards reserving a certain spot in the mosque: First: If he was actually occupying that spot: sitting there, reciting Quran, praying or reading a certain book, but he had to leave it for a certain reason, such as renewing his ablution, answering the call of nature and the like. In his case, he has the right to reserve that spot and nobody is allowed to take it for he is most deserving of it. Rather, if he found someone else there, he is entitled to ask him to leave. This is based on the Hadith in which Abu Hurairah (May Allah be pleased with him) reported: Messenger of Allah (PBUH) said: "If someone leaves his seat (for one reason or another) and First: If he was actually occupying that spot: sitting there, reciting Quran, praying or reading a certain book, but he had to leave it for a certain reason, such as renewing his ablution, answering the call of nature and the like. In his case, he has the right to reserve that spot and nobody is allowed to take it for he is most deserving of it. Rather, if he found someone else there, he is entitled to ask him to leave. This is based on the Hadith in which Abu Hurairah (May Allah be pleased with him) reported: Messenger of Allah (PBUH) said: "If someone leaves his seat (for one reason or another) and returns to it, he is better entitled to it." [Muslim].
Imam Al-Nawawi (May Allah have mercy on him) said: "Our fellow scholars have said: "This Hadith is about the one who took a certain spot in the mosque to perform prayer or recite the Quran or else, but left it for a reason (Renewing ablution, answering the call of nature and the like), then that spot is his. If he returns to it, he is better entitled to it." {Shareh Muslim, vol.14/pp.161}. Second: If he wasn`t actually occupying that spot, but it is a habit of his to pray or recite the Quran there, then he isn`t entitled to it nor permitted to ask whoever took it first to leave. If this could lead to dispute with other worshipers, then this person incurs sin since the mosque is the house of God and none has the authority to appropriate it. Second: If he wasn`t actually occupying that spot, but it is a habit of his to pray or recite the Quran there, then he isn`t entitled to it nor permitted to ask whoever took it first to leave. If this could lead to dispute with other worshippers, then this person incurs sin since the mosque is the house of God and none has the authority to appropriate it. However, scholars who teach Quran or deliver religious lessons at a certain spot of the mosque are excluded from the second case, because such people are more entitled to those spots in all circumstances, and they may ask whoever is occupying them to leave in order to give them room to pass knowledge to learners. Imam An-Nawawi said: "Our fellow scholars have excluded scholars who deliver Fatwas and teach Sharia sciences. These people are more entitled to their spots and may ask those who take them first to leave." {Shareh Muslim, vol.14/pp.160}.
Nonetheless, as for one who is reciting Quran for himself or offering any other act of worship, such person isn`t excluded from the second case and he may not ask anyone to leave the spot he used to occupy unless case one applies to him. For further elaboration on this topic, kindly refer to {Nehayat Al-Mohtaj, vol.5/pp. 346-348}. And Allah Knows Best.