  In this paper we show that the Cutkosky cutting rules are still valid term by
term in the expansion in powers of $\kappa$ of the $\kappa$-deformed 1-loop
correction to the propagator. We first present a general argument which relates
each term in the expansion to a non-deformed amplitude containing additional
propagators with mass $M>\kappa$. We then show the same thing more
pragmatically, by reducing the singularity structure of the coefficients in the
expansion of the $\kappa$-deformed amplitude, to the singularity structure of
non-deformed loop amplitudes, by using algebraic and analytic identities. We
will explicitly show this up to second order in $1/\kappa$, but the technique
can be generalized to higher orders in $1/\kappa$. Both the abstract and the
more direct approach easily generalize to different deformed theories. We will
then compute the full imaginary part of the $\kappa$-deformed 1-loop correction
to the propagator in a specific model, up to second order in the expansion in
$1/\kappa$, highlighting the usefulness of the approach for the phenomenology
of deformed models. This explicitly confirms previous qualitative arguments
concerning the behaviour of the decay width of unstable particles in the
considered model.
