Preheat the oven to 180ºC/gas mark 4, and grease and line the base of a 23cm round springform cake tin.
Pour the orange juice into a measuring jug and add enough cold water to make it up to 275ml. Add to a pan with the apricots. Bring to the boil, then simmer for 10 minutes, until the apricots are soft and pulpy. Cool, then blend until smooth in a food processor.
Put the remaining butter, the caster sugar and most of the orange zest in a bowl and beat together with an electric hand whisk. Gradually whisk in the eggs. Stir in the apricot and orange purée, then fold in the almonds, polenta and baking powder.
Spoon into the prepared cake tin and bake for 45-55 minutes until golden and a skewer inserted into the centre comes out clean. Cool for 10 minutes before turning out on to a wire rack to cool completely. Peel off the baking paper when cooled.
Put 100g butter, the icing sugar and vanilla in a bowl and beat together with an electric hand whisk until smooth. Spread the buttercream over the top of the cold cake and decorate with the remaining orange zest.