If one adds a sheet to his s'chach, the sukkah is invalid. He may have put the sheet above the s'chach to protect from excessive sun. Or, he may have put the sheet beneath the s'chach because of the falling leaves. The sukkah is invalid because now the sheet serves as a covering; the s'chach serves no purpose - and is considered non-existent.
If he smartly used the grapevine or ivy and, without cutting the branches, lifted that and added them to his s'chach, it is invalid - because the sukkah should be made from the "leftover of the threshing floor and winepress." He can cut the branches now, and it will be valid.
The rule: s'chach should be made of material that cannot become ritually impure. Therefore, it cannot be made of food, utensils, or clothing. The s'chach material must also grow from the soil.
Art: Cottage with Peasant Coming Home by Van Gogh