Preheat the oven to 180°C (350°F/Gas 4). Lightly grease a 20cm (8inch) square cake tin and line with baking paper.
Put the banana, pineapple and sugar in a large bowl. Add the sifted flour and cinnamon or mixed spice. Stir together with a wooden spoon until well combined.
Whisk together the oil, pineapple juice and eggs. Pour onto the banana mixture and stir until combined and the mixture is smooth.
Spoon into the tin and smooth the surface. Bake for 1 hour, or until a skewer inserted into the centre of the cake comes out clean. Leave in the tin for 15 minutes, then turn out onto a wire rack to cool.
To make the icing, beat the butter and cream cheese using electric beaters until smooth. Gradually add the icing sugar alternately with the lemon juice. Beat until thick and creamy.
Spread the icing thickly over the top of the cooled cake, or thinly over the top and side.