C

To determine the minimum horizontal velocity \( v \) required for the disc to leave the hemisphere, we first analyze the forces acting on the disc as it moves along the surface of the hemisphere.

1. **Centripetal Force Requirement:** For the disc to stay in circular motion, the centripetal force must be provided by the component of gravitational force acting on the disc at the point of leaving the hemisphere.

2. At the point of leaving, we can set the height of the disc as \( h = R \cos \theta \) and the radius as \( R \), where \( \theta \) is the angle from the vertical. The centripetal force is given by \( \frac{mv^2}{R} \), and the gravitational force component acting down the surface of the hemisphere is \( mg \cos \theta \).

3. Setting these equal for the point just before leaving gives us:
$$ \frac{mv^2}{R} = mg \cos \theta $$
4. Simplifying, we have:
$$ v^2 = gR \cos \theta $$

5. To leave the hemisphere, we require that the normal force becomes zero, hence the minimum condition is at \( \theta = 0 \), where \( \cos 0 = 1 \).

6. Plugging this into our equation gives:
$$ v^2 = gR $$
$$ v = \sqrt{gR} $$

Thus, the smallest horizontal velocity \( v \) that should be given to the disc for it to leave the hemisphere is given by Option C: \( \sqrt{gR} \).