  Basing on the relativistic theory in superstrong magnetic fields (SMFs), we
investigate the influence of strong electron screening (SES) on the rates of
nuclear reaction $^{23}$Mg $(p, \gamma)$ $^{24}$Al by three models of Lai (LD),
Fushiki et al. (FGP), and Liu et al. (LJ) on the surface of magnetars. Our
results show that the rates can be greatly enhanced by three orders of
magnitude due to the influence of SES. The rates in our model are in good
agreement with those of LD and FGP at relatively low density environment (e.g.
$\rho_7<0.01$) for $1<B_{12}<10^2$. On the other hand, in relatively high
magnetic fields (e.g. $B_{12}>10^2$), the rates of our model can be 1.58 times
and around three orders of magnitude larger than those of FGP and LD,
respectively. The significant increase of the rates of our model for $^{23}$Mg
$(p, \gamma)$ $^{24}$Al implies that more $^{23}$Mg will escape from the Ne-Na
cycle due to SES in SMFs. As a consequence, the next reaction $^{24}$Al
$(\beta^+, \nu)$ $^{24}$Mg will produce more $^{24}$Mg to participate in the
Mg-Al cycle. Thus, it may lead to synthesize a large amount of production of
$A\geq 20$ nuclides (e.g. $^{26}$Al) on the surface of magnetars. These heavy
elements (e.g. $^{26}$Al) may be thrown out due to the compact binary mergers
of double neutron star (NS-NS) or black hole and neutron star (BH and NS)
systems. Our results may help to understand why the $^{26}$Al is always
overabundance in the interstellar space. Our conclusion may be helpful to the
investigation of the nucleosynthesis of some heavy elements, the energy
generation rate, and the numerical calculations of magnetars evolution.
