procedure TForm1.Button1Click(Sender: TObject);
var
  Index: Integer;
  CurPos: TPoint;
  NewItem: TMenuItem;
begin
  NewItem := TMenuItem.Create(PopupMenu1);
  Index := PopupMenu1.Items.Count;
  PopupMenu1.Items.Add(NewItem);
  NewItem.Caption := 'Menu Item ' + IntToStr(Index);
  NewItem.Tag := Index;
  NewItem.OnClick := PopupMenuItemsClick;
  // store the current screen relative cursor position
  CurPos := Mouse.CursorPos;
  // invoke the menu to popup at the stored screen position
  PopupMenu1.Popup(CurPos.X, CurPos.Y);
end;