  Employing the standard hard-scattering approach (HSA) in conjunction with the
running-coupling (RC) method, the latter joined with the infrared-renormalon
calculus, we compute power-suppressed corrections $\sim 1/Q^{2n}, n=1, 2,...$
to the massless $\eta^{\prime}$ meson - virtual gluon transition form factor
(FF) $Q^2F_{\eta ^{\prime}g^{*}g^{*}}(Q^2,\omega)$. Contributions to the form
factor from the quark and gluon components of the $\eta^{\prime}$ meson are
taken into account. Analytic expressions for the FF's
$F_{\eta^{\prime}gg^{*}}(Q^2,\omega=\pm 1)$ and
$F_{\eta^{\prime}g^{*}g^{*}}(Q^2,\omega=0)$ are also presented, as well as
Borel transforms $B[Q^{2}F_{\eta^{\prime}g^{*}g^{*}}](u)$ and resummed
expressions. It is shown that except for $\omega=\pm 1, 0$, the Borel transform
contains an infinite number of infrared renormalon poles. It is demonstrated
that in the explored range of the total gluon virtuality $1 \rm {GeV}^{2} \leq
Q^2 \leq 25 \rm {GeV}^{2}$, power corrections found with the RC method
considerably enhance the FF $F_{\eta^{\prime}g^{*}g^{*}}(Q^{2}, \omega)$
relative to results obtained only in the context of the standard HSA with a
``frozen'' coupling.
