This app was mentioned in 6 comments, with an average of 5.33 upvotes
http://online-go.com/ has correspondence game support. I've heard it also works well on phones.
Dragon Go Server is definitely what you're describing. They have native support on iOS and an unofficial client for Android. The website itself isn't so bad on mobile, either:
Android: https://play.google.com/store/apps/details?id=com.hg.anDGS&hl=en
iOS: https://itunes.apple.com/us/app/dragon-go-client/id384372078?mt=8
Website: http://www.dragongoserver.net/
Have fun!
I'd recommend DGS for people looking to play, turn based as you go work out so well on mobile.
https://play.google.com/store/apps/details?id=com.hg.anDGS&hl=en
I used the above one before. Here is a good list.
https://play.google.com/store/apps/details?id=com.hg.anDGS should fit your needs. You'll have to make an account on dragon go server first (free)
anDGS for playing on the Dragon Go Server is really good.
Dragon Go Server is a very good way to play Go asynchronously.