Additive character sum over intersection of additive and multiplicative subgroups of finite fields

Question

Let $H$ be a multiplicative subgroup of the finite field $\mathbb{F}_q$ with $q$ elements, say $H$ is the subgroup of $d$-th powers, $d \mid q-1$. Let $L$ be a subspace of $\mathbb{F}_q$ over some base, say $\mathbb{F}_p$ where $p = \text{char}(\mathbb{F}_q)$. Let $\psi$ be a non-trivial additive character of $\mathbb{F}_q$. Assuming that $H, L, H \cap L$, are not trivial, do we know some good bounds for $$ |S| = \left|\sum_{x \in H \cap L} \psi(x)\right|? $$ Thanks.