I do not capture video myself, but I can tell you that if you specifically want to mute the music, but keep everything else after you have recorded the gameplay, then the only way to do it is if the Hauppauge capture device records each audio channel separately. Then you can edit / mute that specific audio channel.
For example, if you play an online game with 4 other people and all of are using headsets to talk to each other, then there would be total of 8 different audio channels (or audio sources) that can be recorded separately assuming the capture software / hardware allow you to do so. In this scenario the audio channels would be as follows:
#1 - In game dialog (if any)
#2 - Sound affects
#3 - Music
#4 - Mic #1 - You
#5 - Mic #2...