Short answers, pulled from the story.
In existentialism, abandonment refers to the condition of humanity existing without a god who created it with a pre-given purpose. Jean-Paul Sartre defined it in his 1946 lecture L'Existentialisme est un humanisme as meaning simply that God does not exist and that the consequences of his absence must be followed all the way to the end.
Martin Heidegger used "abandonment" as a favored term before Sartre adopted it. Sartre credited Heidegger with the word when he defined it publicly in his 1946 lecture L'Existentialisme est un humanisme, where he gave it the specific meaning of the absence of God.
Sartre identified Christian belief, Christian existentialism, and atheist existentialism. Christian belief holds that God creates people with a purpose; Christian existentialism holds that individuals create meaning through their search for union with God; atheist existentialism holds that there is no human nature because there is no creator.
Heidegger wrote about the abandonment of being rather than abandonment by God, theorizing that this state causes "the distress of lack of distress" and that confronting one's meaningless existence is the truest condition of human being. He identified three concealments of this abandonment: calculation, acceleration, and the outbreak of massiveness.
Murray v. Curlett was the case that removed reverential Bible reading and recitation of the Lord's Prayer from public schools. Sartre cited the atheist petitioners' statement in that case as an expression of the foundational philosophy of abandonment, in which individuals must find in themselves the strength to meet life rather than seeking help through prayer.
Soren Kierkegaard and Friedrich Nietzsche are identified as the supposed originators of existentialist thought. Albert Camus, Michel Foucault, and Noam Chomsky are among the thinkers Sartre associated with the broader atheist orientation that underpins abandonment.