  Motivated by the recent work of Chu-Lee-Tam on the nefness of canonical line
bundle for compact K\"{a}hler manifolds with nonpositive $k$-Ricci curvature,
we consider a natural notion of {\em almost nonpositive $k$-Ricci curvature},
which is weaker than the existence of a K\"{a}hler metric with nonpositive
$k$-Ricci curvature. When $k=1$, this is just the {\em almost nonpositive
holomorphic sectional curvature} introduced by Zhang. We firstly give a lower
bound for the existence time of the twisted K\"{a}hler-Ricci flow when there
exists a K\"{a}hler metric with $k$-Ricci curvature bounded from above by a
positive constant. As an application, we prove that a compact K\"{a}hler
manifold of almost nonpositive $k$-Ricci curvature must have nef canonical line
bundle.
