As depicted in the figure, a concave mirror of radius R is placed...

For the final image to form on the object itself, the light rays must retrace their path after reflection from the concave mirror.

This happens when the rays strike the mirror normally, which means the refracted rays must appear to come from the center of curvature $C$ of the mirror.

The center of curvature $C$ is at a distance $R$ from the pole of the mirror. Since the water is filled up to a height $h$, the distance of $C$ above the water surface is $R - h$.

Let the object be placed at a height $x$ above the water surface. When viewed from the water (denser medium), the apparent height of the object due to refraction at the plane surface is given by:
Apparent height $= \mu x$

For the rays to appear to come from $C$, this apparent height must be equal to the distance of $C$ from the water surface:
$\mu x = R - h$

$x = \dfrac{R - h}{\mu}$

Thus, the object should be placed at a distance of $\dfrac{R - h}{\mu}$ above the water surface.