This offer for those with a Kindle Fire with Special Offers only.
On your Kindle Fire with Special Offers (2nd Generation or HD), swipe the top menu to the left until you see Offers, then press/click in order to see the current offers. At the bottom, you should see the offer to Get $4 off any digital music album. Click on the offer, then on the orange button to add the credit directly to your account. It’s nearly instant and the next screen offers to take you to Amazon MP3 store so you can start shopping. You can also shop from your PC or the Music App on any Android device and the credit will automatically be used on the next qualifying purchase (I prefer to browse on my PC, as you can do a “play all” on the samples).
Offer is good for one album and only one per account (if you have multiple KSO devices, they need to be registered to different accounts when you apply the credit to any of the accounts). The credit must be claimed by November 12 and used by midnight (PT) November 19.
Other than picking out an album that has a price of $7.99 or more, there doesn’t seem to be any restriction on which albums you can choose. The credit will also go onto your account as soon as it is claimed, so you will want to shop for your album right away. I would not wait too long to claim it though, as I’ve had offers disappear before the claim date. Also, if you don’t see the offer at all, try turning off parental controls on your Fire (you can turn them on again, after claiming the offer).