Butter a 1.2litre/2 pint pudding basin. Line the base of the basin with a small piece of baking paper.
Scatter 150g/5½oz of the blackberries into the bottom of the basin and set aside.
Beat the butter, sugar and lemon zest in a mixing bowl until light and fluffy. Whisk half the beaten eggs into the butter mixture. Whisk in half the flour, then the remaining eggs and then the remaining flour. Stir the apple pieces into the mixture. Spoon into the pudding basin and smooth the surface.
Cover the basin with the circle of baking parchment, with the pleat in the centre of the pudding. Cover the parchment with the circle of aluminium foil, again with the pleat in the centre. Tie the pudding very tightly around the rim with string. Create a carrying handle by tying the excess string across the top of the basin and tying it under the string on the opposite side – this will help you lift the pudding out of the pan once it’s cooked. Trim any excess baking paper and aluminium foil, leaving a 2.5cm/1in border, and turn the edges in on themselves to seal.
Put an upturned heatproof saucer or small trivet in a large, deep saucepan, and place the pudding basin on top. Add enough just-boiled water to the pan to come three-quarters of the way up the sides of the basin. Cover the pan with a tight-fitting lid and place over a low heat. Allow to steam in the gently simmering water for 1½-2 hours (depending on how you prefer your pudding cooked). Add more water to the pan if necessary to make sure the pan does not boil dry.
The pudding is done when a skewer inserted into the centre of the pudding (through the aluminium foil and baking paper) comes out clean. If the pudding is still uncooked, cook for a further 30 minutes. When done, turn off the heat and carefully lift the basin from the water. Leave to stand for 5 minutes.
Cut the string from the basin and discard the aluminium foil and paper. Run a table knife around the edge of the pudding to loosen the sides, carefully invert onto a deep plate and remove the basin.
While the pudding is steaming, make the sauce. Heat the jam, remaining blackberries and lemon juice over a low heat for 6-8 minutes, until the jam has melted and the blackberries soften. Remove the pan from the heat. Strain the sauce through a fine sieve into a clean saucepan. Add a little water if the sauce seems too thick and add lemon juice to taste.
Spoon a little of the sauce over the pudding and allow to run slowly down the sides.