It is a fairly special property for the Yoneda embedding $A \hookrightarrow \mathcal{P}A$ of a category to have a left adjoint defined everywhere (this happens just when $A$ is total). However, a partially defined such left adjoint can express the existence of any colimits in $A$. For instance, if $F\colon I\to A$ is any functor, then colimits of $F$ are the same as values of such a partially defined left adjoint at the presheaf $\mathrm{colim}_i\; A(-,F(i))$. A similar idea works for weighted limits in an enriched setting. In fact, $A$ is small-cocomplete just when its Yoneda embedding has a left adjoint defined at all *small presheaves* (presheaves that are small colimits of representables).

My question is, given a presheaf $X\in \mathcal{P}A$, what is a name for the value of a (partially defined) left adjoint to the Yoneda embedding at $X$?

It is tempting to want to call it the "colimit" of $X$, except that $X$ itself is a functor $A^{op} \to \mathrm{Set}$, and we are certainly not talking about the colimit of that functor. If it helps, note that the object in question is equivalently the colimit of the identity functor of $A$ weighted by $X$.