I thought about this a bit when trying to understand $\lambda$-rings – the same example that Wraith offers – and I think the answer is basically what is remarked in the cited lecture notes: it is necessary and sufficient for the new unary operations to distribute over the existing operations in a coherent fashion.
I state a precise theorem below.
For an algebraic theory $T$, let $U^T : \textbf{Mod} (T) \to \textbf{Set}$ be the forgetful functor and let $F_T$ be its left adjoint.
To avoid confusion, and to align with the notation, we speak of $T$-models rather than $T$-algebras or $T$-modules (which are reserved for more complicated things in Wraith's terminology; see below).
For a morphism $f : S \to T$ of algebraic theories, let $U^f : \textbf{Mod} (T) \to \textbf{Mod} (S)$ be the forgetful functor and let $F_f$ be its left adjoint; following the lecture notes, say $f$ is essential if $U^f$ has a right adjoint $H_f$.
(I am avoiding the star notation.)
Recall that coproducts in $\textbf{Mod} (S)$ can be constructed by taking the free $S$-model generated by the disjoint union of the underlying sets of the summands and the quotienting by the evident congruence generated by the "multiplication tables" of the summands; somewhat more precisely, given $S$-models $M_\alpha$ ($0 \le \alpha < \kappa$), the counit gives us a canonical effective regular epimorphism:
$$F_S \left( \sum_{0 \le \alpha < \kappa} U^S (M_\alpha) \right) \cong \sum_{0 \le \alpha < \kappa} F_S U^S (M_\alpha) \twoheadrightarrow \sum_{0 \le \alpha < \kappa} M_\alpha$$
Of course, we may make the same argument for any ad hoc choice of generators for each $M_\alpha$ to get a smaller generating set for $\sum_\alpha M_\alpha$.
It follows that if $U^f$ preserves coproducts of free $T$-models – e.g. if $f$ is essential – then all $\kappa$-ary $T$-operations can be expressed as (the image under $f$ of) an $S$-operation applied to $\kappa$-many unary $T$-operations:
$$U^T F^T (\kappa) \cong U^T \left( \sum_\kappa F^T (1) \right) \cong U^S U^f \left( \sum_\kappa F^T (1) \right) \cong U^S \left( \sum_\kappa U^f F^T (1) \right)$$
This demonstrates $f$ is an extension of $S$ by unary operations that satisfy a distributive law over $S$-operations, at least in the informal sense that any unary $T$-operation applied to any $S$-operation can be rewritten as an $S$-operation applied to some unary $T$-operations.
Let me try to be more precise.
Recall that (for any algebraic theory $T$) $F_T (1)$ admits a canonical internal co-$T$-model structure: indeed,
$$U^T (B) \cong \textrm{Hom}_T (F_T (1), B)$$
and $U^T (B)$ obviously has a natural $T$-model structure, so $F_T (1)$ acquires a co-$T$-model structure (by general Yoneda yoga, i.e. chasing the identity).
This generalises: when $f$ is essential,
$$U^T H_f (A) \cong \textrm{Hom}_T (F_T (1), H_f (A)) \cong \textrm{Hom}_S (U^f F_T (1), A)$$
so $U^f F_T (1)$ acquires the structure of an internal co-$T$-model in $\textbf{Mod} (S)$.
(The earlier version is the special case where $S = T$ and $f = \textrm{id}_T$.)
Furthermore, since $U^f$ preserves coproducts, and both structures are transported from the same source, the structure on $U^f F_T (1)$ can be identified with (the image under $U^f$ of) the structure on $F_T (1)$.
Of course, this is also compatible with the structure on $F_S (1)$ in the sense that the adjunction unit $F_S (1) \to U^f F_T (1)$ is an internal co-$S$-homomorphism (where we are using the obvious analogue of $U^f$ to regard $U^f F_T (1)$ as an internal co-$S$-model).
We have thus proved half of the following:
Theorem.
$f$ is essential if and only if there is an internal co-$T$-model $C$ in $\textbf{Mod} (S)$ with the following properties:
(The underlying object of) $C$ is $U^f F_T (1)$.
For every $\kappa$ and every $t : F_T (1) \to F_T (\kappa)$ in $\textbf{Mod} (T)$, the morphism $U^f (t)$ factors as the given structure morphism $t_C : U^f F_T (1) \to \sum_\kappa U^f F_T (1)$ followed by the canonical morphism $\sum_\kappa U^f F_T (1) \to U^f F_T (\kappa)$ in $\textbf{Mod} (S)$.
The adjunction unit $F_S (1) \to U^f F_T (1)$ (in $\textbf{Mod} (S)$) is an internal co-$S$-homomorphism (where we are using the analogue of $U^f$ to regard $C$ as an internal co-$S$-model).
The "if" direction remains to be proven.
Assume we have such an internal co-$T$-model $C$ in $\textbf{Mod} (S)$.
By Yoneda, we see that $\sum_\kappa U^f F_T (1) \to U^f F_T (\kappa)$ is surjective, so (as discussed earlier) every $\kappa$-ary $T$-operation can be rewritten as an $S$-operation applied to $\kappa$-many unary $T$-operations; the hypotheses of the theorem are (implicitly) the coherence conditions we need to make this work properly.
Indeed, the structure on $C$ means we may define a functor $H^C : \textbf{Mod} (S) \to \textbf{Mod} (T)$ by setting:
$$H^C (A) = \textrm{Hom}_S (C, A)$$
(The RHS acquires a $T$-model structure because $\textrm{Hom}_S (-, A) : \textbf{Mod} (S)^\textrm{op} \to \textbf{Set}$ preserves limits.)
We will show that $H^C$ is a right adjoint of $U^f$ by constructing a unit and counit.
The action of the functor $U^f$ gives a natural $T$-homomorphism $B \to H^C U^f (B)$, by the hypothesis on the factorisation of $U^f (t)$ for each $t : F_T (1) \to F_T (\kappa)$:
$$B \cong \textrm{Hom}_T (F_T (1), B) \to \textrm{Hom}_S (U^f F_T (1), U^f (B)) = H^C U^f (B)$$
Furthermore, we have a natural $S$-homomorphism $U^f H^C (A) \to A$, by the hypothesis on the adjunction unit $F_S (1) \to U^f F_T (1)$:
$$U^f H^C (A) = U^f \textrm{Hom}_S (U^f F_T (1), A) \to \textrm{Hom}_S (F_S (1), A) \cong A$$
These are the desired unit and counit: the triangle identities basically reduce to the fact that $F_T \cong F_f F_S$ and $U^T = U^S U^f$ as adjoints.
Thus, $f$ is indeed essential.
Note that $C$ in the theorem is (among other things) simultaneously an $S$-model and an internal co-$T$-model in $\textbf{Mod} (S)$.
Wraith calls such a thing an $(T, S)$-bimodel.
By Yoneda and the adjoint lifting theorem, the category of $(T, S)$-bimodels is equivalent to the category of left adjoint functors $L : \textbf{Mod} (T) \to \textbf{Mod} (S)$; the $(T, S)$-bimodel corresponding to $L$ is $L F_T (1)$.
In particular, we may define a tensor product of bimodels by identifying it with functor composition.
This makes the category of $(S, S)$-bimodels a (non-symmetric) monoidal category, and Wraith defines an $S$-algebra to be a monoid in this monoidal category.
This, of course, is simply a monad structure on the corresponding endofunctor on $\textbf{Mod} (S)$; a module over an $S$-algebra is simply an algebra for the corresponding monad on $\textbf{Mod} (S)$.
Since the underlying endofunctor of such a monad preserves colimits, the (crude) monadicity theorem can be applied to deduce that the the obvious forgetful functor from the category of modules to $\textbf{Set}$ is monadic, so we obtain an algebraic theory $T$ and an essential morphism $f : S \to T$.
It is also more or less immediate that, for every $T$ and every essential morphism $f : S \to T$, there is corresponding $S$-algebra such that $\textbf{Mod} (T)$ is equivalent to the category of modules over that $S$-algebra as categories over $\textbf{Mod} (S)$.
What the theorem above gives is, in a sense, halfway between these two characterisations.
