Prepare the Oven and Tin: Preheat the oven to 160ºC and line a 2 lb loaf tin with baking paper.
Mix the Batter: In a large bowl, cream together the softened butter, vegetable oil, and caster sugar until the mixture is light, fluffy, and pale in color.
Add Eggs: Beat in the eggs one at a time, ensuring each is fully incorporated before adding the next. Scrape down the sides of the bowl as needed.
Combine Dry Ingredients: Sift in the plain flour and baking powder, then add the orange zest. Gently fold the mixture until just combined, being careful not to overmix.
Bake the Cake: Pour the batter into the prepared loaf tin and bake for 30-40 minutes, or until the top is golden and a toothpick inserted into the center comes out with moist crumbs.
Prepare Soaking Syrup: While the cake bakes, whisk together the orange juice and caster sugar in a bowl until the sugar is dissolved.
Soak the Cake: When the cake is done, leave it in the tin and poke holes all over the top with a toothpick. Pour the soaking syrup evenly over the cake, then cover and let it soak for about 2 hours while the cake cools.
Cool and Ice: Once the cake has soaked and cooled, remove it from the tin and place it on a cooling rack. Whisk together the icing sugar, orange juice, and yoghurt to make the icing. Drizzle the icing over the cake and let it set for about 30 minutes until firm to the touch.