Cover the apricots with boiling water and leave them to plump up for at least 30 minutes. Drain them and reserve 4-5 tablespoons of the liquid.
Peel, deseed and chop the pumpkin. Steam it for 15-20 minutes until very tender. Drain it thoroughly and mash to make 450 ml of purée then set aside for 20 minutes to cool.
Purée the apricots in a food processor or with a hand-held mixer until very smooth, adding the reserved water if necessary to make 125 ml of purée.
Heat the oven to 180°C and grease two 23 cm (9 in) square cake tins.
Place the flour, baking powder, bicarbonate of soda, cinnamon, salt and sugar in a large mixing bowl and make a well in the centre. Add the eggs, oil, apricot and pumpkin purées and mix until they are just combined.
Pour the mixture into the cake tins, smooth the tops and bake for 30-35 minutes, rotating the tins half-way through, until the cakes have risen and a skewer comes out clean.
Cut each cake into 12. Leave to cool in the tins on wire racks for at least 20 minutes, then turn them out onto the racks to cool completely.