Twilio employee here.
TwHello Inc already answered in comments, but just to clarify:
Yes, you can use the <Play>
tag to play back an mp3 file, which could be a recorded voice:
https://www.twilio.com/docs/api/twiml/play
It's as simple as this:
<Response>
<Play>http://mywebsite.com/mysong.mp3</Play>
</Response>