HEre's the breakdown: lps (things with tentacles or just mouths) plus the softies (ranging from mushrooms, buttons, to the leathers) all feed from the water column.
SPS have very tiny mouths, but they only eat particulate the size of bristleworm poo-particles, leading one expert to suggest they have a relationship to the worms. Whatever the truth of this---they mostly feed off light and don't like particulate clouding up their water, for that reason.
I can't speak to softies, but I know that a diet of mysis for the fish and crushed jumbo dried krill, plus calcium-feeding, has enabled wild growth of hammer coral in my tank, with a fairly weak to non-functioning skimmer and strong lighting.
SPS and LPS MUST MUST MUST be fed calcium constantly. A kalk drip can do this in a tank under 75 gallons. Above that, ultimately a calcium reactor is necessary.
Best not to combine these corals unless you're experienced, because a tiny tank can't be all things all at once. Experimenting around to find out what your tank does best is sensible---just don't kill a coral: if you see it's not thriving and you've done what you can, try to trade it back for one you succeed at.