  We discuss central-exclusive production (CEP) of photons via different fusion
processes in the reaction $pp \to pp \gamma$ at high energies, available at
RHIC and LHC, within the tensor-pomeron model. We consider two types of
processes, the photoproduction contribution via the photon-pomeron and
photon-reggeon fusion reactions, and the purely diffractive contribution via
the reggeon-pomeron and odderon-pomeron fusion reactions. We present
predictions for the measurements of photons at midrapidity, $|{\rm y}| < 2.5$,
and at relatively low transverse momentum, $0.1~{\rm GeV} < k_{\perp} < 1~{\rm
GeV}$. To check the main results of our study the measurement of the outgoing
protons is not necessary. This is of relevance, e.g., for the present version
of the ALICE detector at the LHC. Several differential distributions, for
instance, in ${\rm y}$, $k_{\perp}$ and $\omega$, the rapidity, the absolute
value of the transverse momentum, and the energy of the photon, respectively,
are presented. We show that the photoproduction is an important process in the
kinematic region specified above. There it gives a much larger cross section
than diffractive bremsstrahlung. This is remarkable as the CEP cross section is
of order $\alpha_{\rm em}^{3}$ whereas the bremsstrahlung one is only of order
$\alpha_{\rm em}$. On the other hand, the soft-photon bremsstrahlung where the
basic $pp \to pp$ reaction is due to strong interaction diffraction is more
important than CEP in the forward rapidity range, $|{\rm y}| > 4$, and/or at
very low $k_{\perp}$. We leave it as a challenge for the planned ALICE 3
experiment at the LHC to study these two contributions to soft photon
production in $pp$ collisions. This could shed new light on the so called
``soft photon puzzle'' in hadron-hadron collisions.
