Let $L(s, \chi)$ denote the Dirichlet $L$-function associated to the character $\chi$.
In his paper A Mean Value Estimate for Real Character Sums, Heath-Brown proves a mean value bound for $L(s,\chi)$ when $\chi$ is a real character. In particular, he shows that $$ \sum_{\substack{\chi \leq Q \\ \chi \text{ real}}} \lvert L(\tfrac{1}{2} + it, \chi) \rvert^4 \ll (Q (1 + \lvert t \rvert) + 1)^{1 + \epsilon}, \tag{1} $$ where we interpret the sum to be over those real characters with conductors less than $Q$.
I am looking for a proof of $(1)$ (or perhaps a second moment, if that's more readily available) when $\chi$ is allowed to be complex. For instance, suppose that $\chi$ is allowed to run over $n$th-power characters of conductor less than $Q$ (by $n$th-power, I mean that $\chi^n(a) = 1$ for $a$ relatively prime to the conductor of $\chi$).
From what I understand, the key reason why Heath-Brown's work doesn't extend immediately is because he employs a large sieve involving quadratic characters in the form $$ \sum_{m \leq M} \lvert \sum_{n \leq N} a(n) \chi_m(n) \rvert^2 \ll (MN)^\epsilon (M + N) \sum_{n \leq N} \lvert a(n)\rvert^2 $$ where the $a(n)$ are general complex numbers. I am unfamiliar with most work involving such sieves, but I would not be surprised if one had managed to produce good bounds for sieves with $n$-th power characters; and from there, one might produce good bounds for moments of $L$-functions with $n$-th power characters as well.
