Let $\mathscr P$ be an operad in the category of vector spaces. An algebra (of the type encoded by $\mathscr P$) on the vector space $V$ is a morphism of operads $\mu:\mathscr P\to End_V$ with $End_V$ the endomorphism operad for $V$. For example, choosing $\mathscr P$ the operad of associative algebras, then $\mu$ is an associative product on $V$.
What about algebraic structures that presupposes some amount of predefined structures for their definition?
For example, consider a Lie-Rinehart algebra $\big((A,\cdot),(V,*),\rho,[\cdot,\cdot]\big)$ where:
$\bullet$ $(A,\cdot)$ is a commutative associative algebra.
$\bullet$ $(V,*)$ is a $A$-module.
$\bullet$ $\rho:V\to Der(A,\cdot)$ is the anchor.
$\bullet$ $[\cdot,\cdot]:V\otimes V\to V$ is the Lie bracket.
Now, what I'm interested in is the (suitable generalisation of the notion of) operad allowing to describe the pair $(\rho,[\cdot,\cdot])$ on the pair $\big((A,\cdot),(V,*)\big)$. That is, I'm not interested in the "operad" of Lie-Rinehart algebras (for which the commutative associative product $\cdot$ and the module structure $*$ are part of the structure) but only on the one describing the pair $(\rho,[\cdot,\cdot])$ assuming $\big((A,\cdot),(V,*)\big)$.
In other words, (if such generalisation of operads exists), the pair $(\rho,[\cdot,\cdot])$ would be described as a morphism $\mathscr P\to End_{\big((A,\cdot),(V,*)\big)}$ i.e. the right-hand side would "know" about $\big((A,\cdot),(V,*)\big)$.
Is there some generalisation available along these (sketchy) lines in the literature?
Thank you for any reference or remark!