Using pairing in Atiyah-Hirzebruch spectral sequence one can show that homology of $BU(n)$ is a free abelian group with basis $\alpha_{k_1}\cdots\alpha_{k_t}$, $k\leqslant n$, where $\alpha_{i} = \big((c_1)^i\big)^* \in H_{2i}(\mathbb C P^\infty)$. Therefore, passing to (co)limits and applying the Thom isomorphism, we have
\begin{equation}
H_*(MU) \cong \mathbb Z[a_1, a_2, \ldots].
\end{equation}

Since polynomial generators come as image of homology classes of $\mathbb C P^\infty$, $\mathfrak A_p^*$-comodule structure of $H_*(MU; \mathbb F_p)$ can be deduced from the one of $H_*(\mathbb C P^\infty; \mathbb F_p)$.

Is there any similar description of $H_*(MSU; \mathbb F_p)$?

**UPD.**

Let $S = \mathbb F_p[x_n| n+1\neq p^t]$.
Define $f\colon H_*(MU;\mathbb F_p) \to S$ by the formula
\begin{equation*}
	f(a_n) =
		\begin{cases}
		x_n;\quad&\text{if $n+1 \neq p^t$;}
		\\
		0;&\text{otherwise.}
		\end{cases}
\end{equation*}
Then the composition
\begin{equation*}
	H_*(MU;\mathbb F_p)
	\xrightarrow{\text{coaction}}
	\mathfrak A_p'\otimes_{\mathbb F_p} H_*(MU;\mathbb F_p)
	\xrightarrow{\mathrm{id}\otimes f}
	\mathfrak A_p'\otimes_{\mathbb F_p} S
\end{equation*}
is an isomorphism of $\mathbb F_p$-algebras and $\mathfrak A_p^*$-comodules.
Here $\mathfrak A_p' = \mathbb F_p[\xi_1, \xi_2, \ldots]$ is the quotient of 
the dual Steenrod algebra $\mathfrak A_p^*$.

The Adams' paper *Primitive Elements in the $K$-theory of $BSU$*
contains a computation of homology $H_*(BSU; \mathbb Z)$.
Namely, $H_*(BSU; \mathbb Z) \cong \mathbb Z[y_2, y_3, \ldots]$, $\deg y_i = 2i$.
And therefore, $H_*(MSU; \mathbb Z) \cong \mathbb Z[Y_2, Y_3, \ldots]$, $\deg Y_i = 2i$.
But still the proof does not describe the embedding
\begin{equation*}
	H_*(MSU; \mathbb Z) \hookrightarrow H_*(MU; \mathbb Z).
\end{equation*}

Are there any explicit formulae for the images of $y_n$ in $H_*(MU)$?
I would be really grateful for something explicit that could be useful
for describing the $\mathfrak A_p^*$-comodule structure of $H_*(MSU;\mathbb 
F_p)$.