  We calculate the twist-4 corrections to the integral of $g_1(x,Q^2)$ in the
framework of QCD sum rules using an interpolating nucleon field which contains
explicitly a gluonic degree of freedom. This information can be used together
with previous calculations of the twist-3 contribution to the second moment of
$g_2(x)$ to estimate the higher-twist corrections to the Ellis-Jaffe and
Bjorken sum rules. We get $f^{(2)}(proton) = -0.037 \pm 0.006$ and
$f^{(2)}(neutron) = -0.013 \pm 0.006$. Numerically our results roughly agree
with those obtained by Balitsky, Braun and Kolesnichenko based on a sum rule
for a simpler current. Our calculations are far more stable as tested within
the sum rule approach but are more sensitive to less well known condensates.
