Well, I don’t usually have left over round bales these days, but a couple of times in the past it has happened. I tried burning bad cores, adding them to burn piles, but it is TOUGH to get them to light… they tend to smoulder for a while, and go out- they are wrapped too tight to let enough oxygen in to burn well, apparently. It can be done, it just took some effort. But, I must say, I had one core that was just really nasty in mid winter, that became apparent when I was unrolling it to feed. Rotten. I roll the round bales out on the ground to feed, we have no mud, and little rainfall, just snow every now and again. So I scooped that central core back up with my tractor bucket, and tossed it over the fence, into deep bush that was in an unused area, choked with underbrush and deadfall. In later years, I cleared this area, and turned it into a small pasture (my specialty- pasture creation from previously trashed areas!). I had forgotten about the old bale core, it was like 6 years later. It was rolled up against a tree stump, semi sheltered and we don’t get a lot of rain, and the core was not noticeable any more, covered with grasses etc. And I turned horses out there to graze. They ate the bale core, 6 years later. No issues, no illness, they LIKED it. They CHOSE to eat it, instead of fresh grass. It was fermented I guess, not rotten. I was horrified, but they were right, it was OK. Our fairly dry environment makes this sort of thing possible here, more so than in wet environments for sure. Your environment may not be the same as mine. Apparently fermented hay is very popular as horse feed in some parts of the world, quite expensive to buy.
We don’t get bad round bale cores any more, since we got a moisture meter to use while baling them. And a bit of experience about how dry the grass has to be to remain good at the core of a round bale.
If your left over round bale core is good hay, not rotten but just damp on the outside, you can scoop it up with your tractor bucket and a cargo strap, and move it, and feed it. If it is too wet to do this and gooey inside with rot, I’d roll it out and let it decompose and turn into soil in your pasture. You can turn it into the soil later with a rotovator if you like, it will act as fertilizer in your field. I normally just harrow my winter pasture area, in spring when my horses move out of there, breaking up manure and any hay dust/leaves etc that are left over from feeding. Works great! Grows back with thick grass in summer. Previous to our ownership, this area was used for wintering cows, feeding hay on the ground just like this, lots of fertile soil here.
For just a few horses, it is a better plan to serve hay off a round bale from a wheelbarrow, peeling off enough at a time to satisfy your horses daily. Less waste doing it this way.