Hyper-Melaninization
Those small black smudges on the Anemonefish are reactions of the fish coming into contact with a marine lifeform that is irritating their skin. The result is the condition known as Hyper-Melaninization. You can find sample photos of fishes with this condition on the Internet. What you may not find is proper guidance about what to do about it.
Talk about irritation! There seems to be some confusion about what the fish is going through. There are hobbyists, which I must admit irritate me, that say, 'Ignore it and it should go away.' That irritates me because the fish is being harmed by the contact it has with some lifeform in the aquarium.
Anemonefishes have a wonderfully thick mucous coating with allows them to handle the stinging tentacles of their host anemone. They rarely have an adverse reaction to their host anemone. However in the confines of a reef aquarium the Anemonefish faces a lot of different lifeforms unnaturally found in a small space. The Anemonefish has a strong instinct to form a relationship with an anemone and lacking this will attempt to find a 'home' in the lap of another lifeform. Even when there is an anemone available, the Anemonefish may still not host it, but try to host with one or more of the other aquarium lifeforms.
The fish isn't equipped to handle the stings of other lifeforms. Natural Selection has produced an Anemonefish that hosts with a narrow range of anemone lifeforms -- not all of which are right for each fish. So when the fish tries to host with another lifeform, one that harms it, the result is Hyper-Melaninization.
The irritating part is this: Hobbyists often pretend this isn't a problem. But the fish is being stung. Does the fish have to die to convince the hobbyist it is being hurt? Wake up! There is a chance that the fish mucous coating will prevail and protect the fish OR there is a chance the fish will learn to keep away from the stinging lifeform OR there is a chance the fish will continue to be harmed (in part by its need to host). Here's the guideline to follow.
1. Note the date when Hyper-Melaninization is first noted;
2. Watch fish carefully, if those black smudges spread to cover over 90% of the entire body (not including fins and eyes), move the fish out or move the offending lifeform out of the aquarium (you can try A.);
3. If 2. doesn't come about (that is the smudges don't spread to cover more than about 90% of the fish), then leave the fish in the aquarium no more than 10 days with the signs of Hyper-Melaninization;
4. At the end of 10 days if the signs of Hyper-Melaninization are not gone totally or not diminished to some marks are faded and barely visible, then move the fish or offending lifeform out of the aquarium (you can try A.);
5. If after 3 weeks from 1. there is still even the smallest of smudge on the fish, then move the fish or the offending lifeform out of the aquarium -- for the rest of their lives.
A. You can try putting the two back together after all signs of Hyper-Melaninization have left the fish, but if Hyper-Melaninization is noted again, give it up and keep the two separated for the rest of their lives.
Remember, those stings were meant to irritate or kill other fishes. They are not something to ignore. They are not something to treat likely. I wish humans could understand what the fish is going through. It is like we were 'bit' by mosquitoes, but we are not allowed to put on repellent or to get away from the mosquitoes. It isn't 'nothing' to that fish.