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Recently, there appeared a significant interest in inverse spectral problemsfor 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-selfadjointSturm--Liouville operator. We study the inverse problem of recovering thepotential $q\in L_2(0, \pi)$ 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 requiresconsideration of eigenvalues multiplicities. We develop an approach based onthe relation between the characteristic function and the coefficients $\{\xi_n\}_{n \ge 1}$ of the potential $q$ by a certain basis. We obtain necessaryand sufficient conditions on the spectrum being asymptotic formulae of aspecial 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 andthe additional data, we obtain a uniqueness theorem and an algorithm.

On recovering non-local perturbation of non-selfadjoint Sturm-Liouville operator