Position 1 rack in center and 1 rack in lowest position of oven, and preheat to 350°F. Coat 8-inch square baking pan with cooking spray.
Whisk together flour, cinnamon, ginger, baking soda, and salt in medium bowl. Purée 1 1/2 cups of mango chunks in blender or food processor until smooth. Transfer purée to bowl, and whisk in 1 cup sugar, oil, and vinegar. Stir in flour mixture, then fold in raisins.
Spread batter in prepared pan. Slide large empty baking sheet onto bottom rack in oven to diffuse heat. Place cake pan with batter on upper rack, and bake 28 to 30 minutes, or until toothpick inserted in center of cake comes out clean. Cool in pan on wire rack.
Bring 1 cup water, remaining 3/4 cup sugar, and cinnamon pieces to a boil in medium saucepan over medium heat. Reduce heat to medium-low, and simmer syrup 8 minutes, or until mixture is reduced to 1 cup. Transfer syrup (with cinnamon pieces) to shallow bowl to cool.
Stir remaining 2 1/2 cups mangoes into syrup. Cut cake into 12 triangles, and serve with compote spooned over top.
