The ending of The Ballad of Songbirds and Snakes shows Snow choosing power over the one person who trusted him, and it leaves open whether Lucy Gray survived. Spoilers follow.
Here is what happens on screen. Snow confesses his betrayal, and Lucy Gray turns against him. He chases her into the forest to keep her quiet. A snake bites him there, and he does not know whether his shot hit her. She vanishes from his life, and the film never confirms whether she lived.
Did Snow ever actually love Lucy Gray? In my opinion, the film lets both readings stand. He risks the arena for her, which is real feeling, but he also treats her as the key to his family's recovery, and he chases her down once she becomes a threat. It could be argued that he loved her as far as his ambition allowed, which is not very far.
Why did Snow betray Lucy Gray? The film shows him betraying Sejanus Plinth and covering it up, and then pursuing Lucy Gray into the forest to keep her quiet. It never states his motive outright, but, in my opinion, the pattern points to a man protecting himself before anyone else.
The open ending matters for the saga. His choices push him toward the ruthless view of power that defines the tyrant played by Donald Sutherland in The Hunger Games (2012). Because Lucy Gray's fate is never settled, the story stays unfinished in a way I find more unsettling than a confirmed death. For the wider order of events, see the Hunger Games timeline.