Suppose $f$ is a cusp form of half integral weight $k$ w.r.t. the group $\Gamma_0(4)$ ($k$ is not very low, can assume $k \ge 11/2$), and $a_n$ is its Fourier coefficient. The Linnik bound says that if $f$ is square-free then we have the estimate $$ |a_n| \ll_\epsilon n^{k/2-2/7+\epsilon} $$

My question is, is there a nontrivial bound for $\left| \sum_{n=N}^{N+H} a_n \right|$? Here nontrivial means better than simply adding up the Linnik bound individually. Also will averaging take care of the square-free issue in the Linnik bound?

An equivalent problem is to estimate $$ \sum_{n=N}^{N+H} \sum_{4|c} \frac{K(m,n,c)}{c} $$

Note: $m$ is assumed to be fixed and is not averaged. To give some order of magnitude, $H$ is supposed to be around $N^{1/5}$, so it is a rather short sum.