  Super-sample covariance (SSC) is the dominant source of statistical error on
large scale structure (LSS) observables for both current and future galaxy
surveys. In this work, we concentrate on the SSC of cluster counts, also known
as sample variance, which is particularly useful for the self-calibration of
the cluster observable-mass relation; our approach can similarly be applied to
other observables, such as galaxy clustering and lensing shear. We first
examined the accuracy of two analytical approximations proposed in the
literature for the flat sky limit, finding that they are accurate at the 15%
and 30-35% level, respectively, for covariances of counts in the same redshift
bin. We then developed a harmonic expansion formalism that allows for the
prediction of SSC in an arbitrary survey mask geometry, such as large sky areas
of current and future surveys. We show analytically and numerically that this
formalism recovers the full sky and flat sky limits present in the literature.
We then present an efficient numerical implementation of the formalism, which
allows fast and easy runs of covariance predictions when the survey mask is
modified. We applied our method to a mask that is broadly similar to the Dark
Energy Survey footprint, finding a non-negligible negative cross-z covariance,
i.e. redshift bins are anti-correlated. We also examined the case of data
removal from holes due to, for example bright stars, quality cuts, or
systematic removals, and find that this does not have noticeable effects on the
structure of the SSC matrix, only rescaling its amplitude by the effective
survey area. These advances enable analytical covariances of LSS observables to
be computed for current and future galaxy surveys, which cover large areas of
the sky where the flat sky approximation fails.
