You can long-press on what’s playing to bring up options to hide the track, the whole album, or the artist. That stuff will automatically be hidden from the future String.
You can leave the app and return to it later and your hidden items will be respected. These settings will persist until the app is fully closed (from the multitasking view). If you decide to quit the app this way, the next time it loads a String, those items will be back in the String.
If you want to permanently hide artists from a String and save it for future shuffling, you can do so via an in-app purchase.