Does the time for the fasting person's prayer to be answered end simply with the entry of Maghrib time, or does it extend until he actually breaks his fast, even if that is delayed beyond the Azan?

It is recommended for the fasting person to supplicate at the time of breaking the fast, whether the breaking of fast is juridical, by the entry of the time for breaking fast, or actual, by eating and drinking.

It appears that the supplication is to be made after breaking the fast, as mentioned in the hadith of Ibn Amr: "Indeed, for the fasting person, at the time of breaking his fast, there is a supplication that is not rejected." And Ibn Umar would say after breaking his fast: "O Allah, the thirst has gone..."

The matter is broad, so it is permissible to supplicate at the advent of the time for breaking fast, or simultaneously with it, or after it.

It must be noted that the Sunnah is to hasten in breaking the fast.