try this
MenuItem item = NavigationMenu.Items[1].ChildItems[0];
item.Parent.ChildItems.Remove(item);
To remove all the items, we are using reverse loop, since the item is removed from the collection
int count = NavigationMenu.Items[1].ChildItems.Count;
for (int i = count-1; i > -1; i--)
{
MenuItem item = NavigationMenu.Items[1].ChildItems[i];
item.Parent.ChildItems.Remove(item);
}