Lots of research is being done in this area and even a simple Google search will bring up a bewildering number of studies; I got more than 28 million hits in 0.3 seconds! I must confess, I only read a dozen of them, most of which were blogs or popular-level summaries of scholarly work, and I didn't engage in the kind of deep analysis of methodology that the topic probably deserves.
Reasonably soon in my reading it became apparent that, whatever your position on this question, you won't have difficulty finding someone who will encourage you to continue to hold it. Confirmation bias is alive and well.
The upshot is that, while listening to music prior to studying may increase levels of cognitive arousal or alertness(simply because it can put you in a good mood), listening to music while studying has a negative effect on performance. Study takes place best in silence.
For those who look to the music as a way of blocking out other noise, there is some evidence that listening to your favourite music has a more negative effect than listening to something you don't like. Presumably this is because you engage more with the music that you like.
It is also worth noting that music with lyrics has a particularly deleterious effect; a brain processing words through their eyes and words through their ears at the same time, has diminished capacity through cognitive overload.
Some students have told me that they "can't" study without listening to music. I don't doubt their experience, but I suspect it has more to do with the fact that they have formed an unhelpful habit that they find it hard to quit, than that the music is actually helping them. My fear is that they have habituated a behaviour that hurts their performance, a bit like getting used to swimming while towing a bucket.
How are habits to be broken? The only way possible is through focused effort, sustained determination and the formation of new habits. As we all know, habits are easier to break earlier in the formative process.
If your child is developing the habit of listening to music while studying, it may be that this discussion is one worth having. Of course, it would be good not to create a 'battle-ground' in doing so ... but that is a discussion for another article.