Butter a 1.2 litre pudding basin and line the base with baking paper.
Scatter 150g of blackberries into the bottom of the basin.
Beat butter, sugar, and lemon zest until light and fluffy.
Whisk in half the eggs, then half the flour, followed by the remaining eggs and flour.
Stir the apple pieces into the batter and spoon into the basin.
Cover with pleated parchment and foil, then tie securely with string.
Place basin on a trivet in a large pot and add boiling water three-quarters up the sides.
Steam for 90 to 120 minutes until a skewer comes out clean.
Remove from heat and let stand for 5 minutes before inverting onto a plate.
Heat jam, remaining blackberries, and lemon juice for 6-8 minutes.
Strain the sauce through a sieve and adjust consistency with water if needed.
Spoon sauce over the pudding and serve.