A bucket of water is suspended from a spring balance. Does the balance reading change when a piece of iron suspended from a string is immersed in the water? When a piece of cork is put in the water?

Yes, the balance reading changes when a piece of iron is suspended from a string is immersed in the water.
When a piece of iron is immersed in the water suspended from a string, it will experience buoyant force will act on the iron. The buoyant force is equal to the weight of the water displaced by the volume of the iron immersed in the water. Thus, the balance reading is equal to the sum of the weight of the water and weight of the water displaced by the volume of the iron immersed. Therefore, the balance reading will change.
When a piece of cork is put in the water, the balance reading will be equal to the sum of the weight of the water and the cork. Therefore, the balance reading will change.