Is there any mention in the Prophetic Sunnah of something called "Fajr charity" (صدقة الفجر)?

The expression "dawn charity" (صدقة الفجر) is not found in the Qur’an or Sunnah. It is likely taken from weak hadiths, such as: "Give charity early, for indeed calamity does not bypass charity," or from the agreed-upon hadith: "There is no day upon which servants awaken but two angels descend, and one of them says: 'O Allah, give recompense to the one who spends,' and the other says: 'O Allah, inflict destruction upon the one who withholds.'" This hadith does not specify charity at dawn but rather informs about the supplication of the two angels for whoever gives charity on that day. It has been reported in other narrations that the two angels also make that supplication at sunset.