Preheat the oven to 180°C (350°F) and grease and line a 20cm (8 inch) round cake tin with baking parchment.
In a large mixing bowl, cream together the softened butter and caster sugar until pale and fluffy using an electric mixer.
Beat in the eggs one at a time, mixing well after each addition.
Sift the self-raising flour and baking powder into the bowl and fold gently into the mixture.
Add the milk and lemon zest, then fold until the batter is smooth and well combined.
Pour the batter into the prepared cake tin and level the surface with a spatula.
Bake in the preheated oven for 35 to 40 minutes, or until a skewer inserted into the center comes out clean.
While the cake is baking, prepare the lemon drizzle by mixing the icing sugar with the juice of 1 lemon in a small bowl until smooth.
Remove the cake from the oven and while still warm, prick the top all over with a skewer or fork.
Pour the lemon drizzle evenly over the warm cake, allowing it to soak in.
Leave the cake to cool completely in the tin before removing and serving.