  Recently, there appeared a significant interest in inverse spectral problems
for non-local operators arising in numerous applications. In the present work,
we consider the operator with frozen argument $ly = -y''(x) + p(x)y(x) +
q(x)y(a),$ which is a non-local perturbation of the non-selfadjoint
Sturm--Liouville operator. We study the inverse problem of recovering the
potential $q\in L_2(0, \pi)$ by the spectrum when the coefficient $p\in L_2(0,
\pi)$ is known. While the previous works were focused only on the case $p=0,$
here we investigate the more difficult non-selfadjoint case, which requires
consideration of eigenvalues multiplicities. We develop an approach based on
the relation between the characteristic function and the coefficients $\{
\xi_n\}_{n \ge 1}$ of the potential $q$ by a certain basis. We obtain necessary
and sufficient conditions on the spectrum being asymptotic formulae of a
special form. They yield that a part of the spectrum does not depend on $q,$
i.e. it is uninformative. For the unique solvability of the inverse problem,
one should supplement the spectrum with a part of the coefficients $ \xi_n,$
being the minimal additional data. For the inverse problem by the spectrum and
the additional data, we obtain a uniqueness theorem and an algorithm.
