In nuclear fusion, protons do not directly fuse with each other to form heavier elements. Instead, in the process known as proton-proton chain reaction, protons undergo a series of nuclear reactions that eventually result in the formation of helium-4 nuclei.
The fusion process involves a release of energy, but this energy does not enter or merge with the protons themselves. Instead, the energy is released in the form of photons (such as gamma rays) and kinetic energy of the resulting particles. The energy released is a result of the conversion of a small fraction of the mass of the participating particles into energy, as described by Einstein's famous equation E=mc².
To give you a simplified overview of the proton-proton chain reaction:
Two protons come close enough for one of them to undergo a rare quantum tunneling event, turning into a neutron while emitting a positron and a neutrino.
The newly formed neutron quickly captures another proton, forming a deuterium nucleus (one proton and one neutron) and releasing a photon.
The deuterium nucleus then fuses with another proton, creating a helium-3 nucleus (two protons and one neutron).
Two helium-3 nuclei collide and fuse to form a beryllium-6 nucleus (four protons and two neutrons).
The beryllium-6 nucleus rapidly decays, splitting back into two helium-4 nuclei (each with two protons and two neutrons) while releasing two protons.
Throughout this series of reactions, energy is released at each step. This energy originates from the conversion of mass into energy, as dictated by Einstein's equation. The released energy is carried away by photons and by the kinetic energy of the resulting particles, including the helium-4 nuclei and protons.
So, to answer your question, the energy is not fused or merged with the protons themselves. Instead, it is released as a result of the nuclear reactions, contributing to the overall energy output of the fusion process.