In my paper, *A variant of Mathias forcing that preserves $\mathsf{ACA}_0$* [Archive for Mathematical Logic 51 (2012), 751–780; [arXiv:1110.6559](http://arxiv.org/abs/1110.6559), [doi:10.1007/s00153-012-0297-4](http://dx.doi.org/10.1007/s00153-012-0297-4)], I show the needed reals for $F_\sigma$-Mathias forcing are precisely the computable ones. In other words, for any non-computable set $C$ in the ground model there is a $F_\sigma$-Mathias generic $A$ that does not compute $C$. Of course, a $F_\sigma$-Mathias generic will be almost contained in or almost disjoint from any ground model set.

The paper actually goes further and is devoted to showing how $F_\sigma$-Mathias forcing preserves $\mathsf{ACA}_0$. Your collection $\mathfrak{X}$ is actually the second-order part of an $\omega$-model of $\mathsf{ACA}_0$. The preservation results of the paper show that if $A$ is $F_\sigma$-Mathias generic, then the arithmetic closure of $\mathfrak{X}\cup\{A\}$ is just the recursive closure of that set. (More precisely, I define the generic extension of the $\omega$-model with second-order part $\mathfrak{X}$ to be the model whose second-order part is the recursive closure of $\mathfrak{X}\cup\{A\}$, so the preservation theorem then says that this recursive closure is actually arithmetically closed whenever $\mathfrak{X}$ is arithmetically closed.) Combined with a slight strengthening of the cone-avoiding result I proved, you get that there for every $C \notin \mathfrak{X}$ there is a $F_\sigma$-Mathias generic $A$ such that the arithmetic closure of $\mathfrak{X}\cup\{A\}$ avoids $C$. This last part requires some technical work beyond what I wrote in the paper but I am willing to help!