Whenever I feed my fish the corals are catching food. If u feed frozen you can see the coral catching and eat in it. I have even seen some of my mushrooms with pellets in there mouths
And what kind of corals? Soft corals will utilize nitrates and phosphates. Most LPS will eat meaty foods like mysis or krill if those are being offered. SPS eat smaller foods like reef frenzy, oyster eggs or something like that, but most corals can get energy from the algaes in their tissues so light plays into the equation also.
Corals vary. Many corals have algae that grow inside if them. These algae are photosynthetic and provide the coral with food. However, even most of the photosynthetic corals will benefit from external food sources. There are corals that do not have associated algae and these require external food. Remember that corals really do not require much energy so they do not require that much food compared to fish.
Me personally. I don't have the time or patience to feed the 100+ corals I have and fend off all the fish, crabs, and shrimp. I do broadcast 1/3 dose of aquavitro fuel twice a week. Also coral amino twice a week and 1/2 dose microvore once a week. I do not shutoff my skimmer, mp40's or return.
No, you do not need to specifically feed these. If you are feeding your fish well, these coral will thrive on the nutrients that are already in the water.