On $p$-adic $L$-functions for ${\text{GSp}}_4\times {\text{GL}}_2$

We use higher Coleman theory to construct a new $p$-adic $L$-function for ${\text{GSp}}_4\times {\text{GL}}_2$. While previous works by the first author, Pilloni, Skinner and Zerbes had considered the $p$-adic variation of classes in the $H^2$ of Shimura varieties for ${\text{GSp}}_4$, in this note we explore the interpolation of classes in the $H^1$, which allows us to access to a different range of weights. Further, we show an interpolation property in terms of complex $L$-values using the algebraicity results established in previous work by the authors.