Make lemon curd (if homemade): In a small saucepan, whisk together lemon juice, zest, sugar, and eggs. Cook over medium heat, stirring constantly until thickened (about 5–7 minutes). Remove from heat and whisk in butter. Cool completely.
Preheat oven: Heat to 350°F (175°C). Grease and flour an 8x8 or 9-inch round baking pan.
Cream butter and sugar: In a large bowl, beat butter and sugar until light and fluffy. Add eggs one at a time, mixing well. Stir in vanilla, lemon zest, and juice.
Mix dry ingredients: In a separate bowl, whisk together flour, baking powder, and salt. Gradually add to wet mixture, alternating with milk, until just combined.
Assemble: Spread half the batter into the pan. Drop spoonfuls of lemon curd over the batter and swirl lightly. Top with remaining batter and another few swirls of curd on top (if desired).
Bake: Bake for 30–35 minutes, or until a toothpick comes out clean from the center. Cool completely.
Glaze: Whisk powdered sugar with lemon juice to desired consistency. Drizzle over cooled cake before serving.