  A Cardy-like regime of the four-dimensional superconformal index has been
shown to be governed by 't Hooft anomalies and to single out a large-$N$ saddle
carrying the Bekenstein-Hawking entropy of dual supersymmetric black holes in
AdS$_5$. For the universal index where no flavour fugacities are turned on,
this correspondence has been improved by matching the first subleading
corrections to the saddle-point action with the four-derivative corrections to
the black hole action in minimal gauged supergravity, as well as the respective
corrected entropies. Here, we extend this match by including flavour
symmetries. We consider five-dimensional gauged supergravity with vector
multiplet and four-derivative couplings, and provide an effective theory
reproducing the 't Hooft anomalies of the R- and flavour symmetries of generic
holographic superconformal field theories at next-to-leading order in the
large-$N$ expansion. Then we focus on a specific model dual to
$\mathbb{C}^3/\mathbb{Z}_\nu$ quiver gauge theories, where the 't Hooft anomaly
coefficients receive simple but sufficiently generic corrections. In this
model, we evaluate the four-derivative corrections to the on-shell action of
the supersymmetric multi-charge black hole, showing agreement with the
flavoured Cardy-like formula from the index. We give a prediction for the
corrected entropy of the supersymmetric black hole and discuss the general
validity of our results. Taking the limit of infinite AdS$_5$ radius, we also
obtain four-derivative corrections to the action and entropy of supersymmetric
asymptotically flat black holes.
