  In [8](arXiv:2111.06159) we introduced the notion of a
k-almost-quasifibration. In this article we update this definition and call it
a k-c-quasifibration. This will help us to relate it to quasifibrations. We
study some basic properties of k-c-quasifibrations. We also generalize a series
of results on quasifibrations ([1]) to k-c-quasifibrations giving criteria for
a map to be a k-c-quasifibration.
