  We prove a cohomological formula for non-critical residues of degree eight
automorphic $L$-functions of $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ in the
spirit of Beilinson conjecture. We rely on the cohomological interpretation of
an automorphic period integral and on the study of Novodvorsky's integral
representation of the $L$-functions.
