With the rack in the middle position, preheat the oven to 180 °C (350 °F). Line the bottom of a 20-cm (8-inch) springform pan with parchment paper.
In a bowl, combine the flour, baking soda, baking powder, salt, and cinnamon. Set aside.
In another bowl, combine the prune purée and yogurt. Set aside.
In another bowl, cream the butter with the oil, brown sugar, and vanilla with an electric mixer. Add the eggs, one at a time, beating until smooth. At low speed, add the dry ingredients alternately with the yogurt mixture. With a spatula, stir in the walnuts and prunes.
Pour the batter into the prepared pan. Bake for about 1 hour or until a toothpick inserted in the centre of the cake comes out clean. Let cool. Unmould and cool completely on a rack.
In a small saucepan or microwave oven, bring all the ingredients to a boil, stirring until the sugar has dissolved. Let cool for about 10 minutes. Pour onto the cooled cake.