  We prove that any holomorphic vector bundle admitting a holomorphic
connection, over a compact K\"ahler Calabi-Yau manifold, also admits a flat
holomorphic connection. This addresses a particular case of a question asked by
Atiyah and generalizes a result previously obtained in \cite{BD} for simply
connected compact K\"ahler Calabi-Yau manifolds. We give some applications of
it in the framework of Cartan geometries and foliated Cartan geometries on
K\"ahler Calabi-Yau manifolds.
