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The effect of explicit chiral-symmetry breaking on inhomogeneous chiral phases is studied within a Nambu-Jona-Lasinio model with nonzero current quark mass. Generalizing an earlier result obtained in the chiral limit, we show within a Ginzburg-Landau analysis that the critical endpoint of the first-order chiral phase boundary between two homogeneous phases gets replaced by a "pseudo-Lifshitz point" when the possibility of inhomogeneous order parameters is considered. Performing a stability analysis we also show that the unstable mode along the phase boundary is in the scalar but not in the pseudoscalar channel, suggesting that modulations which contain pseudoscalar condensates, like a generalized dual chiral density wave, are disfavored against purely scalar ones. Numerically we find that the inhomogeneous phase shrinks as one moves away from the chiral limit, but survives even at significantly large values of the current quark mass., Comment: 9 pages, 4 figures; v2: reference and minor typos corrected; v3: references added, comment on fluctuation effects, version accepted for publication at Phys. Lett. B