Atomic excitation via photon absorption

Hi guys! I have come across a problem I can't seem to wrap my head around. I've learned that E.M. radiation can be propagated by discrete packets of energy , photons. Is the energy of each photon discrete, or can it have a continuous range of energies depending on its frequency? I would be inclined towards the latter, though. Then what does discrete actually mean in terms of energy propagation? For an atom to be excited, it has to absorb a photon whose energy is exactly equal to the energy needed for a quantum jump. Why can't the atom absorb what energy it needs from the photon(assuming that the photon has more energy than the atom needs) and leave the photon with the remainder? Whereas for the photoelectric effect, any photon above the threshold energy can be absorbed? Thank you for reading, have a great day.