Twisted quadratic Kloosterman-sum bound with additive twists

Let $a,q\geq2$ be integers with $(a,q)=1$ and $q$ not a perfect square. Let $H,K\geq1$ and $\lambda,\mu$ be reals. Define $e(u):=e^{2\pi i u}$, let $\left(\frac{\cdot}{\cdot}\right)$ denote the Jacobi symbol, and let $\bar h$ denote the multiplicative inverse of $h$ modulo $q$. Twisted quadratic Kloosterman-sum bound with additive twists. For any $\epsilon>0$,

$$\begin{aligned} &\mathop{\sum_{1\leq h\leq H}}_{(h,q)=1}e(\lambda h)\sum_{0\leq k<K}e(\mu k)\left(\frac{h}{q}\right)e\left(\frac{a\bar h k^2}{q}\right)\\ &\ll_\epsilon\left(H^{1/2}K^{1/2}+H^{3/4}+K+q^{-1/2}HK+q^{-1/2}K^2\right)q^\epsilon. \end{aligned}$$

This is proposed as the stronger estimate needed when the phase has additive twists; the source gives no resolution evidence.