What is the ruling on one who prays "Tahiyyat al-Masjid" (greeting the mosque prayer) while the obligatory prayer is being established? And how does one cut off a supererogatory prayer if the obligatory prayer is established?

If the prayer is announced, it is not permissible for a Muslim to begin a supererogatory (nafilah) prayer, due to the hadith: "When the prayer is announced, there is no prayer except the obligatory (maktubah) one." This prohibition is considered absolute and حرمة (harâm), whether one fears missing a rak'ah or not, because the default understanding of a prohibition is that it signifies unlawfulness, and because "no prayer" encompasses every situation, even for someone who is in the midst of a supererogatory prayer, in which case they must cut it short. The supererogatory prayer can be cut short by saying the taslim, or by any other action that invalidates it, or simply by turning away.