In category theory the adjective “weak” is used when the uniqueness parts of a universal property is removed.
For example, a weak product of $A$ and $B$ is an object $P$ with morphisms $p_1 : P \to A$ to $p_2 : P \to B$, such that, for all $f : C \to A$ and $g : C \to B$ there exists (not necessarily unique!) $h : C \to P$ satisfying $p_1 \circ h = f$ and $p_2 \circ h = g$.
We may apply the same idea to exponentials. A weak exponential of $A$ and $B$ is an object $E$ with a morphism $e : E \times A \to B$ such that, for all $f : C \times A \to B$ there exists (not necessarily unique!) $h : C \to E$ such that $e \circ (h \times \mathrm{id}_A) = h$. This is just the usual universal property of exponentials, with the uniqueness part removed.
An example of a category which has weak exponentials but no exponentials is the category of topological spaces and continuous maps. In more syntactic settings, such as logical frameworks, removing the uniqueness part from the definition of exponentials corresponds to removing the $\eta$-rule. Consequently, given a term $z : C, x : A \vdash f : B$, there may be many terms $z : C \vdash h : A \to B$ satisfying $$z : C, x : A \vdash f \equiv h\, x : B.$$
To see that the $\eta$-rule gives uniqueness, given $h_1$ and $h_2$ satisfying the above equation, observe that $$h_1 \equiv (\lambda x \,. h_1 \, x) \equiv (\lambda x \,.f) \equiv (\lambda x \,. h_2 \, x) \equiv h_2.$$
Conversely, from uniqueness we get $\eta$-rule as follows: given any $z : C \vdash h : A \to B$, observe that
$$z : C, x : A \vdash h\,x \equiv h\,x : B$$
and by $\beta$-reduction
$$z : C, x : A \vdash h\,x \equiv (\lambda y \,. h\,y) \, x : B$$
therefore by uniqueness $z : C \vdash h \equiv (\lambda y \,. h\,y) : A \to B$.
As an exercise you can verify that weakness of products correspond to absence of the $\eta$-rule $(\pi_1\,t, \pi_2\,t) \equiv t : A \times B$.
Let us also try to answer the question “what does it have to do with HOAS?“ Recall that in HOAS we use the meta-level function space to model binding in the object-level terms. Does the meta-level $\eta$-rule transfer to the object-level $\eta$-rule? For that we would need something like (I am using $\mathtt{typeface}$ to denote object-level syntactic constructs):
$$\mathtt{lambda}(\lambda x \,. \mathtt{app}(h,x)) \equiv h,$$
which is not the case because “object-level application $\mathtt{app}$ is not meta-level application“. So in HOAS the object-level $\eta$-rule is not available, hence “weak” exponentials.
Incidentally, the meta-level $\beta$-reduction fails to transfer to object-level $\beta$-reduction also, because the object-level $\beta$-reduction is something like
$$\mathtt{app}(\mathtt{lambda}(t), u) \equiv t \, u,$$
and once again the equation does not hold. Of course, one could consider HOAS with object-level equations, in which case both the $\beta$-reduction and the $\eta$-rule could be imposed.
