More than one recent scientific study has determined that shoaling/schooling fish need a group of close to 10 or more. One study tested the species in individual groups of three, five and ten. The groups of three and five had increased aggression, the groups of ten did not. The smaller groups also displayed a latency to feed. And for fish to be reluctant to feed, it shows they are significantly affected.
I believe you had an earlier thread on pH in which you mentioned the 2 neons left from the original 10. Replacing the group is obviously one option, or leaving the two and going with another species. Sometimes when one, two, or three are left, as when the others naturally die off from old age after a few years, the remaining fish may or may not show signs of stress from this. I have had it go both ways. I had one Red Phantom Tetra left from the original 15 and this last fish was in the tank for 3 years before it too died. On the other hand, my surviving five pencilfish from a group of close to 20 for many years became real terrors, tearing into each other and the rummynose, so I euthanized them.
Given the aggression obvious here, one neon will soon be on its own. But it may also attack any other fish, for the same reason. If this were me, with this small tank, I would euthanize the neons and move on. I know it is not easy, but you absolutely cannot have any aggressive fish bullying others, it is severely stressful and they will then themselves succumb to this or that. And, when any fish has developed symptoms of aggression, it is never reversible because the effect has harmed the fish itself.
As for the cory, there is something but I never guess with disease issues, so hopefully other members will be able to pin this down. It could be an attack from the neon, or disease.
@DoubleDutch @GaryE