My first suggestion would be to do a significant water change now, at least half the tank volume but more up to 70% advisable. Us a conditioner, but nothing else.
This is clearly a problem respirating, but it could be any one of several things: lack of oxygen, ammonia, nitrite, pH shock, toxin in the water. More data on this tank will possibly help us narrow this down. How long has it been running, was it cycled.
What is the pH? And the GH? The white gravel substrate, do you know what it is composed of? White can sometimes be calcareous, dissolving mineral into the water, which would not help neon tetra (or cories).
Cories often go the the surface for a gulp of air, but this should not be too frequent; the more they do it, the more likely something is wrong in the water, same issues as above.