  Sums of Kloosterman sums have deep connections with the theory of modular
forms, and their estimation has many important consequences. Kuznetsov used his
famous trace formula and got a power-saving estimate with respect to $x$ with
implied constants depending on $m$ and $n$. Recently, in 2009, Sarnak and
Tsimerman obtained a bound uniformly in $x$, $m$ and $n$.
  The generalized Kloosterman sums are defined with multiplier systems and on
congruence subgroups. Goldfeld and Sarnak bounded sums of them with main terms
corresponding to exceptional eigenvalues of the hyperbolic Laplacian. Their
error term is a power of $x$ with implied constants depending on all the other
factors. In this paper, for a wide class of half-integral weight multiplier
systems, we get the same bound with the error term uniformly in $x$, $m$ and
$n$.
  Such uniform bounds have great applications. For the eta-multiplier, Ahlgren
and Andersen obtained a uniform and power-saving bound with respect to $m$ and
$n$, which resulted in a convergent error estimate on the Rademacher exact
formula of the partition function $p(n)$. We also establish a Rademacher-type
exact formula for the difference of partitions of rank modulo $3$, which allows
us to apply our power-saving estimate to the tail of the formula for a
convergent error bound.
