Let $\mathcal{A},\mathcal{B}$ and $\mathcal{C}$ be three categories. Let $F:\mathcal{A}\to \mathcal{B}$ be a functor, and let $G\dashv H$ be an adjunction, where $G:\mathcal{B}\to\mathcal{C}$ and $H:\mathcal{C}\to\mathcal{B}$.
If $G\circ F$ is to have a right adjoint, is it necessary that
i) there exists a functor $K:\mathcal{B}\to\mathcal{A}$? And, if yes,
ii) is $K$ right adjoint to $F$?
i) Your assumptions imply that all three categories are either empty or nonempty simultaneously, so that there certainly exists some functor $B\to A.$ But that's not very interesting.
ii) There's no reason $F$ should have a right adjoint. For instance, if $A,C$ are both copies of the category with a single object and arrow, and if $B$ is a category with a terminal object, then your assumptions all hold. But $F$ will only have a right adjoint if it happens to send the object of $A$ to an initial object of $B$.