At the Mass I attend, soft music is played in the background when the priest is praying the eucharistic prayers. Is this permitted?
No. The General Instruction of the Roman Missal says the following:
The nature of the presidential texts demands that they be spoken in a loud and clear voice and that everyone present listen with attention. While the priest is speaking these texts, there should be no other prayer or liturgical song, and the organ or other instruments should not be played. (GIRM 32)