Agnostic Learning of K-Juntas over "small" distribution
Uniform convergence bounds should imply that agnostic learning with respect to a succinct distribution specified by a finite set S is no easier than learning with respect to the uniform distribution, at least for large enough $S$. Specifically, for any class of functions $C$ with VC-dimension $d$, the canonical algorithm which takes a randomly sampled collection of $m$ points for $m = O(d/\epsilon^2)$ and finds the function $f \in C$ which minimizes error with respect to the sample, will be within an additive $\epsilon$ of the minimum error function over the entire target distribution. So if you have an algorithm for agnostically learning with respect to an explicitly represented set of points $S$, you can learn over the uniform distribution by just taking a sufficiently large sample of points $S$ and feeding $S$ to your algorithm for concise distributions. In fact, this holds for any distribution, not just the uniform distribution.

Now there are ${d \choose k}\cdot 2^{2^{k}}$ size $k$ juntas, and so they have VC-dimension at most $d \leq 2^k + k\log d \approx 2^k$.  So for large enough $S$ (at least, larger than $2^k$), lower bounds for agnostically learning w.r.t. the uniform distribution (or any other distribution) should apply to the problem of learning w.r.t. a concise distribution.