  Sums of the singular series constants that appear in the Hardy--Littlewood
$k$-tuples conjectures have long been studied in connection to the distribution
of primes. We study constrained sums of singular series, where the sum is taken
over sets whose elements are specified modulo $r$ or weighted by smooth
functions. We show that the value of the sum is governed by incidences modulo
$r$ of elements of the set in the case of arithmetic progressions and by
pairings of the smooth functions in the case of weights. These sums shed light
on sums of singular series in other formats.
