Is a land sale considered legitimate, where its price was agreed upon and paid, then an excess part was discovered to have been sold mistakenly, so the buyer paid its price to the seller, and subsequently the buyer paid an amount to the seller's children who claimed their father had promised them the land?

The action taken regarding the sale of the extra piece is valid, and it is considered a new sale. It was not incumbent upon the questioner to pay anything to the seller's two sons, and they do not have the right to demand money or to annul the sale because of their father's promise. The original sale was valid even before that, and if the questioner willingly gives them any money, then there is no harm.