No, she will not take the eggs back if you remove them from the nest. The eggs will go bad and you will have a nasty explosive mess in your house. If there is till one egg left in the nest, she will go back to the nest and lay the remaining clutch for this season. If you remove the eggs, she will lay more eggs to replace the ones you took as the mother swan can count. This is called double clutching.

Female swans are programmed to lay a certain amount of eggs throughout their lifetime and through each breeding season. Double clutching can be detrimental to the female because she loses more than the usual 30% during initial nesting and can possibly increase egg impaction which can be fatal.

We would definitely add straw and build up the nest before she lays more eggs. You will need to especially concentrate on shoring up the bottom of the nesting (egg) bowl. The Regal Swan