  In this paper we revisit the problem of identifying bona fide cluster
Cepheids by performing an all-sky search for Cepheids associated with open
clusters and making use of state-of-the-art catalogued information for both
Cepheids and clusters, based on the unparalleled astrometric precision of the
second and early third data releases of the Gaia satellite. We determine
membership probabilities by following a Bayesian approach using spatial and
kinematic information of the potential cluster-Cepheid pairs. We confirm 19
Cepheid-cluster associations considered in previous studies as bona-fide, and
question the established cluster membership of six other associations. In
addition, we identify 138 cluster Cepheid candidates of potential interest,
mostly in recently discovered open clusters. We report on at least two new
clusters possibly hosting more than one Cepheid. Furthermore, we explore the
feasibility of using open clusters hosting Cepheids to empirically determine
the Cepheid period-age relation through the use of Gaia and 2MASS photometry
and a semi-automated method to derive cluster ages. We conclude that the usage
of cluster Cepheids as tentative probes of the period-age relations still faces
difficulties due to the sparsely populated red giant branch and the
stochastically sampled main-sequence turn-off of the open clusters, making age
determinations a challenging task. This biases the age-dateable cluster
selection for Cepheid period-age studies towards older and high-mass clusters.
