A metal cube is placed inside an empty vessel. When water is added...

Initially, the force exerted by the metal cube on the bottom of the empty vessel is equal to its weight, $W$.

When water is added and the cube is completely immersed, the water exerts pressure on the exposed surfaces of the cube. Since the bottom of the cube is in perfect contact with the bottom of the vessel, no water enters between them, and thus there is no upward buoyant force acting on the bottom surface of the cube.

However, the water above the cube exerts a downward force on its top surface due to liquid pressure. Let this downward force be $F = h \rho g A$, where $h$ is the depth of the water above the cube, $\rho$ is the density of water, and $A$ is the area of the top surface.

The total downward force acting on the cube, which is balanced by the normal reaction from the bottom of the vessel, is now $W + F$.

Since the new force $W + F$ is greater than the initial force $W$, the force on the bottom of the vessel in contact with the cube will increase.