Very hard question, and probably something that any musician must have felt at some point in their musical journey. Yet I haven't heard it being discussed that much.
I do not pretend to have an answer to this at all, but I figured I'd share some thoughts on this based on my subjective experience to complete the other interesting existing answers.
Your question relates to your own subjective enjoyment of your music. The concept of Shoshin ("Beginner's mind") in Zen Buddhism is what pops to my mind when reading your question. According to wikipedia :
[Shoshin] refers to having an attitude of openness, eagerness, and lack of preconceptions when [approaching an activity].
According to this article,
The moment something becomes familiar to us, the mind tries to take over, calling up memories, and creating expectations and fears which stop us from connecting directly to the reality of the present moment.
My guess would be that a part of the joy we have when performing a new song comes from this openness and unexpecting attitude. From my experience, meditating can make me get closer to this state of openness, and it also usually makes me enjoy my own (and any) music more.
On the same note, the best "course of action" might actually just be to accept it. It's a common thing, familiarity can strip a piece of his magic. It might be that the expection of that "punch", of being moved by the piece is one of the reasons you don't feel it anymore.
Be open and ready to be or not be moved, and let any emotion or lack of emotion flow. Enjoyment of music is in nature a very spiritual activity, therefore we don't have any recipe to reach any emotion 100% of the time. Let it flow :)