$x^{2} y\left(\frac{1}{x^{2}}+\frac{2}{x y}\right)$

Step 1 ：Simplify the given expression: $x^{2} y\left(\frac{1}{x^{2}}+\frac{2}{x y}\right)$

Step 2 ：Combine the terms inside the parentheses: $x^{2} y\left(x^{-2}+\frac{2}{x y}\right)$

Step 3 ：Distribute $x^{2}y$ to the terms inside the parentheses: $x^{2}y(x^{-2}) + x^{2}y\left(\frac{2}{x y}\right)$

Step 4 ：Simplify the terms: $y + 2x$

Step 5 ：\(\boxed{2x + y}\) is the final simplified expression.