I encounter a tricky sum like the Kloosterman sum

$${\sum_{x \mod P}}^\ast e\left(\frac{ax+\overline{x}}{P}+\frac{lx^2}{P^2}\right),$$ where $l$ is a positive integer co-prime with $P$ and here $P$ is a prime number, $a$ is a positive integer co-prime with $P$. $\overline{x}$ means $x\overline{x}≡1(\mod P)$.

If this sum can be bounded by sqauare-root $O(P^{\frac{1}{2}+\varepsilon})$? (which follows by invoking Weil bound for Kloosterman sums or some knowledge on exponential sums estimates developed by some experts like A. Adolphson, S. Sperber, E. Bombieri, N. Katz which part I am not familiar with, to be frankly) Did anyone ever saw this kind of sum before with a square module $P^2$ at the tail? Please share some comments. Many thanks.

Note that one has the estimate for the exponential sum: for $(a,p)=1$, and $\chi$ a character modulo $P$ $${\sum_{x \mod P}}^\ast \chi(x)e\left(\frac{ax+\overline{x}}{P}\right)\ll P^{\frac{1}{2}}.$$ In our case the exponential factor is of modulo $P^2$. By now I haven't fond any reference using the sophisticated knowledge on exponential sum in finite field to get a power-saving bound (noting the completing method would give the trivial bound $O(P)$). If any expert here have some strategies or references, please give a guide.

Your any opinions are highly appreciated. Thanks in advance.