If $A\dashv B\colon \cal H\to K$, then the condition $AB$ has right adjoint $T$ is clearly equivalent to the condition that:

$$\hom(A(B(-)), =) \approx \hom(B(-), B(=))$$

is representable. You may actually think of $\hom(B(-), B(=))$ as of the canonical "promonad" (a monad in the bicategory of profunctors) associated to $B$. Such a promonad is representable if and only if $B$ (as a functor) has an absolute right lifting along itself.

Therefore, up to the fact that $B$ has left adjoint, your question may be rephrased as: "when does a functor, which has an absolute right lifting along itself, have a right adjoint?". I doubt there are general conditions (unless trivial) to answer this question.

If $A\dashv B\colon \cal H\to K$, then the condition $AB$ has right adjoint $T$ is clearly equivalent to the condition that:

$$\hom(A(B(-)), =) \approx \hom(B(-), B(=))$$

is representable. You may actually think of $\hom(B(-), B(=))$ as of the canonical "promonad" (a monad in the bicategory of profunctors) associated to $B$. Such a promonad is representable if and only if $B$ (as a functor) has an absolute right lifting along itself (this is a form of op-density condition).

Therefore, up to the fact that $B$ has left adjoint, your question may be rephrased as: "when does a functor, which has an absolute right lifting along itself, have a right adjoint?". I doubt there are general conditions (unless trivial) to answer this question.