  We build a one-variable $p$-adic $L$-function attached to two Hida families
of ordinary $p$-stabilised newforms $\mathbf{f}$, $\mathbf{g}$, interpolating
the algebraic part of the central values of the complex $L$-series $L(f \otimes
\textrm{Ad}(g), s)$ when $f$ and $g$ range over the classical specialisations
of $\mathbf{f}$, $\mathbf{g}$ on a suitable line of the weight space. The
construction rests on two major results: an explicit formula for the relevant
complex central $L$-values, and the existence of non-trivial $\Lambda$-adic
Shintani liftings and Saito--Kurokawa liftings studied in a previous work by
the authors. We also illustrate that, under an appropriate sign assumption,
this $p$-adic $L$-function arises as a factor of a triple product $p$-adic
$L$-function attached to $\mathbf{f}$, $\mathbf{g}$, and $\mathbf{g}$.
