Motivic cohomology and infinitesimal group schemes

For $k$ a perfect field of characteristic $p>0$ and $G\mathrm{/}k$ a split reductive group with $p$ a non-torsion prime for $G,$ we compute the mod $p$ motivic cohomology of the geometric classifying space $B{G}_{(r)}$, where $G_{(r)}$ is the $r$th Frobenius kernel of $G\mathrm{.}$ Our main tool is a motivic version of the Eilenberg-Moore spectral sequence, due to Krishna. For a flat affine group scheme $G\mathrm{/}k$ of finite type, we define a cycle class map from the mod $p$ motivic cohomology of the classifying space $BG$ to the mod $p$ 'etale motivic cohomology of the classifying stack $\mathcal{B}G\mathrm{.}$ This also gives a cycle class map into the Hodge cohomology of $\mathcal{B}G\mathrm{.}$ We study the cycle class map for some examples, including Frobenius kernels.