  The goal of this note is to study the spectrum of a self-adjoint convolution
operator in $L^2(\mathbb R^d)$ with an integrable kernel that is perturbed by
an essentially bounded real-valued potential tending to zero at infinity. We
show that the essential spectrum of such operator is the union of the spectrum
of the convolution operator and of the essential range of the potential. Then
we provide several sufficient conditions for the existence of a countable
sequence of discrete eigenvalues. For operators having non-connected essential
spectrum we give sufficient conditions for the existence of discrete
eigenvalues in the corresponding spectral gaps.
