I encounter a tricky sum like the Kloosterman sum

$$\sum_{x \mod qP} e ( \frac{x+\overline{x+P}}{qP} ),$$ where $q$ is a positive integer, $P$ is a prime number satisfying $(q,P)=1$, $x \bmod qP,(x,qP)=1,$ $ \overline{x} $ means $ x\overline{x} \equiv 1\pmod {qP}$.

If this sum can be bounded by $O((qP)^{1/2+\varepsilon}$) by invoking Weil bound for Kloosterman sums? Did anyone ever saw this kind of sum before?Please share some comments. Many thanks.

My confusions are as follows: (a) One may try to split the sum into two sums with the modulos being $q$ and $P$. However we may have the issue that $y+1$ is not co-prime with $q$ when writing $qP=xq+yP$ with $x \bmod P,(x,P)=1,$ and $y \bmod q,(y,q)=1.$ So that it seems that one cannot write $\overline{xq+(y+1)P}\mod {qP}$ as the form $x\cdot A+y\cdot B$ for some integers $A,B$.

(b) For typical modulos, for example, to consider the sum $$\sum_{x \mod c} e ( \frac{x+\overline{x+P}}{c} ),$$ where $c$ is an arbitrary positive integer. If we have the square-root cancellation for this type of sum, just like the Wiel bound?

Your any opinions are highly appreciated. Thanks in advance.