Could anyone please tell me what is responsible for these little white eggs, similar to ants eggs, I have found in the roots of the broad beans as I was clearing them down.
They are nodules that form on the roots of leguminous plants such as beans. They fix nitrogen from the air, so that the plant can use it. They will cause no harm. You can either leave them to rot in the ground, or aff to the compost heap.
In fact they are a good thing as they can put nitrogen back in the ground. Don't remove broad bean roots too early. You can just cut the plants right down and leave the roots.