Hi Kate & welcome,
Firstly, if he's suffering heel &/or sole abscesses, I suggest he may not be going so well barefoot as you think, as it could be due to stone bruising. Therefore, I'd be sticking to yielding footing or protecting his feet with boots or such in situations that are problematic for him, for the time being at least. Abscesses at the coronet band, if not from the odd injury there, can signify different things & it's possible, for eg, if he's worked on hard ground & hoof walls are bearing the whole/majority of the load, that it's concussion related.
So... diet & nutrition are indeed vital for health, including hoof health & strength. Low carb feed is important, so grain & other starchy ingredients should be avoided, and sugary, cattle fattening 'improved' pastures(such as ryegrass for eg) can also be a problem. Nutrition-wise, a basically well balanced diet is a good start. Extra Magnesium & omega 3s are a couple of specifics big on the normal balancing list. Gravelproofhoof.org is one source to learn more about nutrition as it relates to hooves.
If you would like a critique of your horse's feet, opinions on specifics, check out the link in my signature & post some pics of his feet, and info on current diet, environment, work, etc.