  We calculate single-logarithmic corrections to the small-$x$ flavor-singlet
helicity evolution equations derived recently in the double-logarithmic
approximation. The new single-logarithmic part of the evolution kernel sums up
powers of $\alpha_s \, \ln (1/x)$, which are an important correction to the
dominant powers of $\alpha_s \, \ln^2 (1/x)$ summed up by the
double-logarithmic kernel at small values of Bjorken $x$ and with $\alpha_s$
the strong coupling constant. The single-logarithmic terms arise separately
from either the longitudinal or transverse momentum integrals. Consequently,
the evolution equations we derive simultaneously include the small-$x$
evolution kernel and the leading-order polarized DGLAP splitting functions. We
further enhance the equations by calculating the running coupling corrections
to the kernel.
