  We prove the Bloch--Kato conjecture for certain critical values of degree 8
$L$-functions associated to cusp forms on $\mathrm{GSp}_4 \times
\mathrm{GL}_2$. We also construct a $p$-adic Eichler--Shimura isomorphism in
Hida families for $\mathrm{GSp}_4$, relating $H^2$ of automorphic vector
bundles with $\mathbf{D}_{\mathrm{cris}}$ of a subquotient of \'etale
cohomology.
