What are the best fully explicit upper bounds one can give for the sum $$\left\lvert \sum_{n=N}^{\infty} \frac{S(a,b;n)}{n} \,I_1\!\left(\frac{4 \pi \sqrt{|ab|}}{n}\right) \right\lvert$$ where $S(a,b;n) = \sum_{\gcd(k,n)=1} e^{2 \pi i (ak+bk')/n}$ is the ordinary Kloosterman sum?
As a generalization one may consider any sufficiently regular function $f(C/n)$ instead of the Bessel function.
A trivial bound follows from using Weil's bound $|S(a,b;n)| \le \tau(n) \sqrt{\gcd(a,b,n)} \sqrt{n}$, plugging in an explicit bound for $\tau(n)$, and summing the series (see for example Brisebarre and Philibert), but this is quite pessimistic.
The problem of bounding such sums is discussed in Sarnak and Tsimerman, On Linnik and Selberg's conjecture about sums of Kloosterman sums. As far as I can tell, there are no published bounds (other than the trivial one) with fully explicit constants.
For motivation, see: https://arxiv.org/abs/2011.14671
