  We have developed an approximate way of dealing with explicit
energy-dependence of non-local nucleon optical potentials as used to predict
the $(d,p)$ cross sections within the adiabatic theory. Within this
approximation, the non-local optical potentials have to be evaluated at an
energy shifted from half the incident deuteron energy by the $n-p$ kinetic
energy averaged over the range of the $n-p$ interaction and then treated as an
energy-independent non-local potential. Thus the evaluation of the distorting
potential in the incident channel is reduced to a problem solved in our
previous work in [{\it Phys. Rev. Lett. 110, 112501(2013) and Phys. Rev. C 87,
064610 (2013)}]. We have demonstrated how our new model works for the case of
$^{16}$O$(d,p)^{17}$O, $^{36}$Ar($d,p)^{37}$Ar and $^{40}$Ca$(d,p)^{41}$Ca
reactions and highlighted the need for a detailed understanding of
energy-dependence of non-local potentials. We have also suggested a simple way
of correcting the $d-A$ effective potentials for non-locality when the
underlying energy-dependent non-local nucleon potentials are unknown but
energy-dependent local phenomenological nucleon potentials are available.
