Preheat the oven to 170°C /Gas 4.
Lightly grease the sides of a loaf tin and line with nonstick baking paper.
In a large mixing bowl, combine the sugar and butter and, using an electric whisk, beat until smooth. Once smooth, add one egg at a time, beating between each one until thoroughly combined.
Next, add the flour, baking powder and allspice to the bowl and whisk again until fully blended, then add the carrot and apple and stir together well.
Pour into the loaf tin and place on the middle shelf of the oven to bake for about 45 minutes. After the first 25 minutes of cooking time you may find you need to cover the cake with foil to prevent it from going too brown on top. To test if the cake is ready, insert a clean wooden skewer into the centre; if the batter is still gooey it will need another 10 minutes or so.
Remove from the oven and set aside to cool for half an hour, then transfer to a plate. The easiest way to do this is by holding a folded clean tea towel in one hand and flipping the cake onto it. Remove the tin and any baking parchment and replace with an upturned plate before carefully flipping the cake the right way up with the plate underneath.
In a small bowl, make your icing.
Mix together the icing sugar and lemon juice until smooth (you may need to add a small dash of warm water if the icing is too thick), then pour the icing all over the cooled cake, allowing it to drip down the sides. You can do this while the cake is still warm.
Leave to cool completely before serving with a good cup of tea.
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