If you ionise hydrogen (strip off the electron) you get a free proton (i.e not hydrogen). If that proton recombines with an electron to become a neutral hydrogen atom again, is it a new hydrogen atom?
Solar flares throw out ionised hydrogen (basically, free protons and electrons)- these eventually slow down and recombine to form hydrogen. The question is, is it new hydrogen, or recycled hydrogen?
Any process that actually generates free protons may well end up generating new hydrogen - if the protons slow down without combining with a nucleus. This is most likely to occur if the protons are low energy.
High energy protons can collide with a nucleus and be absorbed
(eg Hydrogen + proton -> Deuterium + emitted positron)
There’s also the process of proton emission where highly excited (say, after beta decay) or very proton rich nuclei spontaneously emit protons, which would become hydrogen atoms after electron capture.
Nuclear fission reactions often produce free neutrons. If those neutrons aren’t absorbed by any other nucleus, they’ll decay into protons (and electrons, and anti-neutrinos), and probably those protons with capture electrons eventually to make hydrogen atoms.
I was astounded to discover that thunderstorms can create photons of enough energy to liberate measurable neutrons. Still, over the history of the planet a rough guesstimation would suggest it would yield about enough hydrogen to allow you make a cup of coffee.
It makes no difference - a free proton that was once part of a hydrogen atom is indistinguishable from one produced by proton emission, and it will behave identically. Both are likely to slow down, find an electron and pair up into elemental hydrogen.
The question of whether the reformed hydrogen atom is the same hydrogen atom or a different hydrogen atom is one for a philosopher, not a physicist.
It my reckoning, it really doesn’t. To a first approximation, the universe is 3/4 Hydrogen and 1/4 Helium, and has been since the universe got cool enough for nuclei to form. The processes that make new free protons (or neutrons, or deuterons) all involve heavier, usually much heavier nuclei. If you took all of the uranium in the universe and broke up every nucleus completely into protons and neutrons, and then let the neutrons decay into protons, you still wouldn’t budge the hydrogen abundance by any noticeable amount.
Of course that takes “hydrogen atom” to mean “hydrogen nucleus”. You take it to mean actual atom such that if a hydrogen atom loses its electron, and then regains another one it is counted as a different atom, I doubt there is very much primordial hydrogen in the universe at all.
Think of it this way: Every proton that wasn’t around and free since the Big Bang has instead come out of some heavier nucleus or another. And each of those heavier nuclei was in turn produced from, ultimately, hydrogen. The vast majority of nuclei don’t emit protons or neutrons at all, and the vast majority of primordial hydrogen has never been fused into anything larger. So the “new protons” must be far, far rarer than the primordial.