Your premise is a Sokoban variant (more generally, a turn-based square grid puzzle based on moving adjacent pieces, usually in order to reach the exit, with frequent opportunities to fail hard, like Kye or Lasertank) featuring a person (very commonplace) and four magical dragons (unusual, and therefore begging to have a central role).
What can the dragons do? What pieces can the player and the dragons interact with? The game itelf, not an arbitrary tacked-on plot, can take many different directions.
- Do you want dragons to be faithful animal helpers against monsters and bandits? Then let the player herd moving enemies into the firebreathing range of his dragons, which could be fixed in the map like turrets. It's the main puzzle type in DRoD.
- Do you want dragons to be a precious thing to transport and protect? Then let them move in stupid and inconvenient ways, with a highly mobile player clearing a way for them and neutralizing traps and hazards. Likely similar to Sokoban (mostly passive obstacles needing accurate rearrangement).
- Do you want dragons to provide magical effects to the protagonist? Then he can pull them out of his shirt pocket as needed, like Pokémon. Puzzles can involve collecting keys and resources (to fuel spells), and roguelike-style powerful enemies (without random combat mechanics) are a possibility.
- Do you want dragons to be creepy? Put innocent bystanders in your level, and allow them to be harmed by the player's dragons (more or less intentionally). For extra cynicism, the player could take advantage of their deaths like in Lemmings (e.g. 2 corpses fill a pit trap).