Weighted fourth moments of Hecke zeta functions with groessencharacters

We use recently obtained bounds for sums of Kloosterman sums to bound the sum $\sum_{-D\leq d\leq D} \int_{-D}^D |\zeta(1/2+it,\lambda^d)|^4| \sum_{0<|\mu|^2\leq M} A(\mu)\lambda^d((\mu)) |\mu|^{-2it}|^2 {\rm d}t$, where $\lambda^d$ is the groessencharacter satisfying $\lambda^d((\alpha)) = \lambda^d(\alpha{\Bbb Z}[i]) = (\alpha /|\alpha|)^{4d}$, for $0\neq\alpha\in{\Bbb Z}[i]$, and $\zeta(s,\lambda^d)$ is the Hecke zeta function that satisfies $\zeta(s,\lambda^d) =(1/4)\sum_{0\neq\alpha\in{\Bbb Z}[i]} \lambda^d((\alpha)) |\alpha|^{-2s}$ for $\Re(s)>1$, while the numbers $D,M\in(0,\infty)$ and function $A:{\Bbb Z}[i]-\{0\}\rightarrow{\Bbb C}$ are arbitrary (though it is only in respect of cases in which $M$ is relatively small, compared to $D$, that our results are new and interesting). One of our new bounds may have an application in enabling a certain improvement of a result of P.A. Lewis on the distribution of Gaussian primes.