Preheat the oven to 180°C (160°C fan/Gas Mark 4). Grease and line a 20cm round baking tin.
In a large bowl, cream together 175g softened unsalted butter and 175g caster sugar until light and fluffy.
Beat in 3 large eggs one at a time, adding a spoonful of 175g plain flour with each egg to prevent curdling.
Gently fold in the remaining plain flour and 1 teaspoon baking powder, then stir in 2 tablespoons milk and 1 teaspoon vanilla extract until just combined.
Pour the batter into the prepared tin and bake for 25–30 minutes, or until a skewer inserted into the centre comes out clean. Cool completely in the tin.
Meanwhile, make the buttercream: beat 100g softened unsalted butter until pale and creamy. Gradually add 200g sieved icing sugar, beating well after each addition.
Stir in ½ teaspoon vanilla extract and then add 1½ tablespoons milk, one at a time, until the buttercream is smooth and spreadable.
Crumble the cooled cake into a large bowl, ensuring no large lumps remain. Add two-thirds of the buttercream and mix with your hands until a dough forms. Add more buttercream if needed.
Roll the mixture into 3–4cm balls and place on a baking tray lined with baking paper. Melt 50g white chocolate and dip the tip of each of 22 lollipop sticks into it; insert halfway into each cake ball. Chill in the fridge for at least 30 minutes.
Melt the remaining 350g white chocolate in a bain-marie or microwave. Dip each chilled cake pop into the chocolate, turning to coat evenly, and tap off excess.
Before the coating sets, sprinkle colourful sprinkles over each cake pop. Stand upright to set fully, then chill for another 15–20 minutes.
Store in an airtight container in the fridge for up to 3–4 days. Bring to room temperature 15–20 minutes before serving.