This is very much a question of recent research, solved with varying degrees of generality in the papers listed below. Here is a summary.
The rational map $\mathbb{P}^{2[8]}\dashrightarrow \mathbb{P}^2$ from the Hilbert scheme of $8$ points in $\mathbb{P}^2$ is the map corresponding to an extremal effective divisor on the Hilbert scheme. This suggests that you should run the minimal model program for the Hilbert scheme, studying the birational modifications that arise on the way from the standard model of the Hilbert scheme until you reach the model where the map to $\mathbb{P}^2$ becomes a morphism. (Note that the Hilbert scheme is a Mori dream space, so this goal is actually realistic.)
Along the way, you'll find that the models you obtain can be realized as moduli spaces of certain Bridgeland semistable objects. In the general case of $n$ points, the Hilbert scheme will be replaced by a Grassmannian bundle over a moduli space of representations of a Kronecker quiver; in your $n=8$ point case this simplifies to a $G(2,9)$-bundle over $\mathbb{P}^2$, as follows.
Consider the space consisting of pairs $(p,\Lambda)$ where $p\in\mathbb{P}^2$ is a point and $$\Lambda\in G(2, H^0(\mathcal{O}_{\mathbb{P}^2}(3) \otimes I_p))$$ is a two-plane in the space of cubics passing through $p$. Thus, this space is a $G(2,9)$-bundle over $\mathbb{P}^2$. It is evidently birational to the Hilbert scheme, since a general $(p,\Lambda)$ determines a length $8$ scheme residual to $p$ in the intersection of the cubics in $\Lambda$. And, the forgetful map to $\mathbb{P}^2$ is your Cayley-Bacharach map.
The references for more general questions of this type are:
Daniele Arcara, Aaron Bertram, Izzet Coskun, and Jack Huizenga, The minimal model program for the Hilbert scheme of points on $\Bbb{P}^2$ and Bridgeland stability, Adv. Math. 235 (2013), 580--626. (Explicitly runs the MMP for $n\leq 9$ points, so your question is a special case.)
Coskun, H., Woolf, "The effective cone of the moduli space of sheaves on the plane."
H., "Effective divisors on the Hilbert scheme of points in the plane and interpolation for stable bundles."
Also there is a survey which might help get you started:
Izzet Coskun and Jack Huizenga, The birational geometry of the moduli spaces of sheaves on $\Bbb P^2$, Proceedings of the Gökova Geometry-Topology Conference 2014 (2015), 114--155.
