  We study leading string corrections to the type IIB supergravity solution
dual to the ${\cal N}=1$ supersymmetric $SU(N+M)\times SU(N)$ gauge theory
coupled to bifundamental chiral superfields $A_i, B_j$, $i,j=1,2$. This
solution was found in hep-th/0007191, and its asymptotic form describing
logarithmic RG flow was constructed in hep-th/0002159. The leading tree-level
string correction to the type IIB string effective action is represented by the
invariant of the form $\a'{}^3 (R^4 + ...)$. Since the background contains
3-form field strengths, we need to know parts of this invariant that depend on
them. By analyzing the 5-point superstring scattering amplitudes we show that
only a few specific $R^3 (H_3)^2$ and $R^3 (F_3)^2$ terms are present in the
effective action.
  Their contribution to the holographic RG flow turns out to be of the same
order as of the $R^4$ terms. Using this fact we show that it is possible to
have agreement between the $\alpha'$-corrected radial dependence of the
supergravity fields and the RG flow dictated by the NSVZ beta functions in
field theory. The agreement with field theory requires that the anomalous
dimension of the operators $\Tr (A_i B_j)$ is corrected by a term of order
$(M/N)^4 \lambda^{-1/2}$ from its value $-{1 \over 2} $ found for M=0
($\lambda$ is the appropriate 't Hooft coupling which is assumed to be strong).
