I take it that you treat the links as tracks (like the songs) and essentially build the whole show in the playout log.
What you are hearing is PlayIt Live crossfading between the items (as you would expect for songs).
Probably the simplest way to deal with this is to use the Seque Editor
If you move the fade envelope on your links that should sort it.
The Segue Editor is extremely powerful and you will have to see what the practical effect is on your program of doing this (also you can adjust the way the songs fade like this).