lol be patient.
On Mac OSX there's a few ways to add songs to iTunes from LW.
What you need to keep in mind is the playlist you see called '(name) LW tunes' or whatever you chose to call it, is simply a playlist. The songs themselves are not in the iTunes music library, but instead are in the LW download folder. And the playlist points iTunes to find them there. If you should remove the songs from the download folder then the playlist won't be able to find them.
Best way is to add the music files to iTunes Music Library.
This can be done automatically using LW's auto import function, however this is not always reliable. Go to LW's preferences (keyboard shortcut is Command - Comma
⌘ ,). Then choose the iTunes tab, then select as shown below. But tick the import option, example below shows it not ticked. To manually add songs, see
How to transfer music to my iPod (& iTunes) {click on link} 
LW Playlist to show in iTunes:
