  Among the best known quantities obtainable from photon correlation
measurements are the $g^{(m)}$~correlation functions. Here, we introduce a new
procedure to evaluate these correlation functions based on higher-order
factorial cumulants $C_{\text{F},m}$ which integrate over the time dependence
of the correlation functions, i.e., summarize the available information at
different time spans. In a systematic manner, the information content of
higher-order correlation functions as well as the distribution of photon
waiting times is taken into account. Our procedure greatly enhances the
sensitivity for probing correlations and, moreover, is robust against a limited
counting efficiency and time resolution in experiment. It can be applied even
in case $g^{(m)}$ is not accessible at short time spans. We use the new
evaluation scheme to analyze the photon emission of a plasmonic cavity coupled
to a single quantum dot. We derive criteria which must hold if the system can
be described by a generic Jaynes-Cummings model. A violation of the criteria
can be explained by the presence of an additional excited quantum dot state.
