What is this prayer performed by this woman in the mosque after the Maghrib prayer, which includes reading from the Mushaf, standing, and a long prostration with a rosary under her belly, and is it permissible to leave the mosque when she performs it?

There is no harm in the time of supererogatory prayers, in the person performing supererogatory prayers reading from the Mushaf, or in prolonging prostration. What this woman is doing in terms of supererogatory prayers does not warrant disapproval, annoyance, or leaving the mosque.