Welcome to the board.
It depends on what you want to have happen. If you PasteSpecial, then the value will be in the cell, but the dropdown still exists. However, the dropdown will not incorporate that new value.
If you want it incorporated, then it is a much more complicated process. You will need to use a ComboBox (Control Toolbar), and you would need to update by including the new value in the drop down using VBA.