Consider the connective $K$-theory spectrum $ku$. Let $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum and $H\mathbb{F}_p$ be the mod-$p$ Eilenberg-MacLane spectrum.
Is it known what $ku^{*}(H\mathbb{Z})$, or $ku^{*}(H\mathbb{F}_{p})$, is?
Consider the connective $K$-theory spectrum $ku$. Let $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum and $H\mathbb{F}_p$ be the mod-$p$ Eilenberg-MacLane spectrum.
Is it known what $ku^{*}(H\mathbb{Z})$, or $ku^{*}(H\mathbb{F}_{p})$, is?
Charles Rezk already answered this in the comments; I'll just expand on what he wrote. This paper discusses what's now known as Mahowald-Rezk duality; this is a version of Anderson duality that takes into account "additive degeneration". I'll briefly recall some facts from that paper before mentioning how to use it to prove Rezk's claim. As in their paper, I'll be $p$-completing everything in sight for some fixed prime $p$.
Let $A^\ast$ (resp. $A_\ast$) denote the mod $p$ (dual) Steenrod algebra. Define a functor $\widetilde{J}$ from left $A_\ast$-comodules to left $A^\ast$-modules via $\widetilde{J}(M) = \mathrm{Hom}_{A_\ast}(A_\ast, M)$, and if $M$ is finitely generated, let $\widetilde{I}(M)$ denote the dual $\widetilde{J}(M)^\vee$. In section 8 of their paper, Mahowald and Rezk show that if $X$ is a fp-spectrum of fp-type $\leq n$ (intuitively, this is the statement that $X$ is in the thick subcategory generated by $\mathrm{BP}\langle n\rangle$; I don't know if being an fp-spectrum of fp-type $\leq n$ is equivalent to this), then there is a spectrum $W_n X$ such that $\mathrm{H}_\ast(W_n X) = \widetilde{I}(\mathrm{H}_\ast X)$. The spectrum $ku$ (again, $p$-completed) is a fp-spectrum of fp-type $\leq 1$, so there is a spectrum $W_1 ku$ such that
$$\mathrm{H}_\ast(W_1 ku) = \widetilde{I}(\mathrm{H}_\ast ku) = \mathrm{Hom}_{A_\ast}(A_\ast, \mathrm{H}_\ast ku)^\vee = [\mathrm{H}\mathbf{F}_p, ku]_\ast^\vee,$$
where the final identification comes from Lemma 6.2 of their paper. It therefore remains to compute $W_1 ku$, which is done in Corollary 9.3 of their paper: $W_1 ku = \Sigma^4 ku$. It follows that $\mathrm{H}^{\ast-4}(ku;\mathbf{F}_p) = ku^\ast(\mathrm{H}\mathbf{F}_p)$. The claim about the $ku$-cohomology of $\mathrm{H}\mathbf{Z}$ can be deduced from this using the cofiber sequence $\mathrm{H}\mathbf{Z} \xrightarrow{p} \mathrm{H}\mathbf{Z} \to \mathrm{H}\mathbf{F}_p$.