Ok, I want to share my experience with Music HD on iOS, guess all the iOS users would make the same findings.
I've been using my iPod Touch 6gen for streaming for years, with a DFR attached to it permanently. This also worked fine with Amazon Music, until they updated the app and I upgraded to Music HD.
HD and SD content still work, when downloaded (I always download first when I play a record, and my listening habits are to listen to a full record).
However, the play time with a fully loaded iPod's battery is around 3hrs for HD content, then the battery dies. In comparison, I'm getting between 5-6 hrs with MQA on Tidal and around 6-7hrs with Spotify.
With Ultra HD content, it's a different story. 24/48 plays fine, but on about a half of 24/96 and all 24/192 tracks I'm getting hickups. The playback buffer goes flat and playback stops until the buffer is populated again. And this is with a playback from the internal flash storage, no streaming. The device is getting VERY HOT to touch. Obviously, the amazon's app is using up whole processor power available, which is still not enough for hickup-free playback. And the battery goes flat in 1-1.5hrs, depending on content.
This is ridiculous, even though it's HiRes, it's still just audio, and iPod Touch 6gen is perfectly capable of playing HiRes Video, not just audio. Amazon's developers really messed up the iOS app.
Now Part 2. Just by a coincident I ran into a nice deal last week and bought a NOS iPhone 7 with 256GB for 270 Euro. It's even by far cheaper than a new iPod Touch 7gen with 256GB, which is sold for 449 Euro here, and still a much better device than an iPod. I intend to use it solely for music with DFR, just replacing the iPod Touch...
Anyway, as expected, the iPhone gives me playback times which are about 2x of those of iPod, as the battery is about x2 in size. And this time, the processor power is enough for hickup-free Ultra HD playback.
BUT: also this device is getting very hot with Ultra HD. And, more severe, Ultra HD playback is not possible without a trick that further reduces already low playback time.
The issue with Ultra HD playback: as soon as the display switches off or the app goes into background because you want to do something else while it's playing, iOS would terminate the Amazon music app within seconds. This only happens with Ultra HD content, not with HD. Probably either memory or power consumption is too high for a background app, so iOS decides to kill it.
Workaround: keep the app always in a foreground, and leave the display always on, so that the app always stays in a foreground and won't be terminated by iOS.
All described issues, with both iPad and iPhone go down to the same actual root cause: this app is a power hog, and is consuming WAY more resources than is appropriate for Audio playback. I already provided this feedback on amazonforum, hopefully Amazon would do something to fix this...