No, the absorption or emission of a photon by a hydrogen atom does not change the atom's mass. The mass of an atom is primarily determined by the combined masses of its protons and neutrons, which are located in the nucleus. Photons, on the other hand, have no rest mass.
When a hydrogen atom absorbs a photon, it undergoes a transition from one energy level to another, resulting in an excited state. The absorption of the photon increases the atom's energy, but it does not change the mass of the atom. The mass of the atom remains constant throughout the process.
Similarly, when a hydrogen atom emits a photon and transitions to a lower energy state, the energy carried away by the photon does not affect the mass of the atom. The total mass of the atom is conserved in such processes.
It's worth noting that while the mass of the atom does not change during absorption or emission of a photon, the momentum of the atom-photon system can be altered. This is due to the transfer of momentum between the atom and the photon. However, this effect is related to momentum rather than mass.