The first moment the chirality-odd twist-3 parton distribution $e(x)$ is related to the pion-nucleon $sigma$-term which is important for phenomenology. However, the possible existence of a singular contribution proportional to $delta(x)$ in the distribution prevents from the determination of the $sigma$-term with $e(x)$ from experiment. There are two approaches to show the existence. The first one is based on an operator identity. The second one is based on a perturbative calculation of a single quark state with finite quark mass. We show that all contributions proportional to $delta (x)$ in the first approach are in fact cancelled. To the second approach we find that $e(x)$ of a multi-parton state with a massless quark has no contribution with $delta (x)$. Considering that a proton is essentially a multi-parton state, the effect of the contribution with $delta(x)$ is expected to be suppressed by light quark masses with arguments from perturbation theory. A detailed discussion about the difference between cut- and uncut diagrams of $e(x)$ is provided.
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