It seems it’s impossible to wake Siri up using the “Hey Siri” functionality while playing music (using either internal speakers and headphones.)
Is there a way to solve this issue?
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As far as I'm aware Hey Siri isn't actually supported while music is playing.
People on Reddit have reported the same issue about a year ago - https://www.reddit.com/r/iphone/comments/7ssnz7/hey_siri_not_working_when_music_is_playing/
On my iPhone SE it works while music is playing.
Try going to Settings -> General -> Software Update and updating your iOS.
You can also try to be nearer the microphone if the music is playing louder.
Also be aware that Hey Siri only works on battery on the iPhone 6S and later so if ypir phone is older than that you need to have it plugged in.
It's also worth checking Siri is on by holding the Home Button.
Tested using iOS 12.1.1