Kloosterman sums and Maass cusp forms of half integral weight for the modular group

We estimate the sums $$\sum_{c\leq x} \frac{S(m,n,c,\chi)}{c},$$ where the $S(m,n,c,\chi)$ are Kloosterman sums of half-integral weight on the modular group. Our estimates are uniform in $m$, $n$, and $x$ in analogy with Sarnak and Tsimerman's improvement of Kuznetsov's bound for the ordinary Kloosterman sums. Among other things this requires us to develop mean value estimates for coefficients of Maass cusp forms of weight $1/2$ and uniform estimates for $K$-Bessel integral transforms. As an application, we obtain an improved estimate for the classical problem of estimating the size of the error term in Rademacher's formula for the partition function $p(n)$.