What you are looking for is the two photon photoelectric effect. In this case, the electron absorbs (part of) the energies of two photons, and gets ejected. Now it is very important to understand the work function of the electron. This is the minimum thermodynamic work to remove an electron from the solid lattice.
The photoelectric work function is the minimum photon energy required to liberate an electron from a substance, in the photoelectric effect. If the photon's energy is greater than the substance's work function, photoelectric emission occurs and the electron is liberated from the surface.
Excess photon energy results in a liberated electron with non-zero kinetic energy.
Now you are asking what happens when all of the energy of one photon is absorbed by one electron, but the electron is still not ejected, even though the photon's energy is above the work function. Or you are asking what happens when only part of the photon's energy is enough to eject the electron.
There are a few cases:
this is QM, all about probabilities, and thought the photon's energy is above the work function of the electron, the electron is still not ejected. In this case part of the photon's energy is transferred to the molecule's vibrational and rotational energies (heating up the material)
two photon absorption, where a single electron absorbs two photon's energy (in part), and gets ejected (because even part of the two photon's energy is above the work function)
Two-photon absorption (TPA) is the absorption of two photons of identical or different frequencies in order to excite a molecule from one state (usually the ground state) to a higher energy, most commonly an excited electronic state.
We report on the observation of two-photon electron emission from silver nanoparticles suspended in nitrogen flow resulting from irradiating them with continuous wave and pulsed laser light with photon energies below the threshold of the single-photon photoelectric effect. The photoelectron yield is quadratic in the light intensity, and the two-photon electron emission threshold is evident. The efficiency of the two-photon photoelectric effect is determined for nanoparticles of various sizes. These experiments offer the net information on nonlinear quantum properties of an isolated single nanoparticle which is crucial for developing theoretical models.