This is a special case of the general construction of cocompletions that preserve existing colimits. The general statement can be found as Theorem 6.23 of Kelly's Basic Concepts of Enriched Category Theory, and more explicitly as Proposition 11.4 and Theorem 11.5 of Fiore's Enrichment and Representation Theorems for Categories of Domains and Continuous Functions (in the case of small cocompletions, though the result is easily modified to work with a class of colimits instead). In summary, for classes $\Phi, \Psi$ of colimits for which the small category $\mathbf B$ is $\Phi$-cocomplete, there is a conservative $\Psi$-cocompletion $\widehat {\mathbf B}_\Phi$ of $\mathbf B$ preserving the $\Phi$-colimits. This means that the restriction of the (restricted) Yoneda embedding $\mathbf B \to \widehat {\mathbf B}_\Phi$ is $\Phi$-cocontinuous and exhibits a bijection between $\Phi$-cocontinuous functors $\mathbf B \to \mathbf C$ into cocomplete categories $\mathbf C$, and $\Phi$- and $\Psi$-cocontinuous functors $\widehat {\mathbf B}_\Phi \to \mathbf C$. Explicitly, $\widehat {\mathbf B}_\Phi$ is the subcategory of the category of presheaves on $\mathbf B$ which are finite colimits of representables taking $\Phi$-cocones to limiting $\Phi$-cones.

In your setting, take $\Phi$ to be the class of colimits in the image of $\iota : \mathbf A \to \mathbf B$, and take $\Psi$ to be the class of finite colimits. Then $\widehat {\mathbf B}_\Phi$ is exactly the finite cocompletion of $\mathbf B$ relative to $\iota$.

This is a special case of the general construction of cocompletions that preserve existing colimits. The general statement can be found as Theorem 6.23 of Kelly's Basic Concepts of Enriched Category Theory, and more explicitly as Proposition 11.4 and Theorem 11.5 of Fiore's Enrichment and Representation Theorems for Categories of Domains and Continuous Functions (in the case of small cocompletions, though the result is easily modified to work with a class of colimits instead). In summary, for classes $\Phi, \Psi$ of colimits for which the small category $\mathbf B$ is $\Phi$-cocomplete, there is a conservative $\Psi$-cocompletion $\widehat {\mathbf B}_\Phi$ of $\mathbf B$ preserving the $\Phi$-colimits. This means that the restriction of the (restricted) Yoneda embedding $\mathbf B \to \widehat {\mathbf B}_\Phi$ is $\Phi$-cocontinuous and exhibits a bijection between $\Phi$-cocontinuous functors $\mathbf B \to \mathbf C$ into cocomplete categories $\mathbf C$, and $\Phi$- and $\Psi$-cocontinuous functors $\widehat {\mathbf B}_\Phi \to \mathbf C$. Explicitly, $\widehat {\mathbf B}_\Phi$ is the subcategory of the category of presheaves on $\mathbf B$ which are $\Psi$-colimits of representables taking $\Phi$-cocones to limiting $\Phi$-cones.

In your setting, take $\Phi$ to be the class of colimits in the image of $\iota : \mathbf A \to \mathbf B$, and take $\Psi$ to be the class of finite colimits. Then $\widehat {\mathbf B}_\Phi$ is exactly the finite cocompletion of $\mathbf B$ relative to $\iota$.