Adding a core will increase the stiffness of the structure for a given amount of fiber. This happens because the laminate gets thicker. The thicker the laminate is the stiffer it becomes. All composite materials “stretch” when they are loaded. When a laminate bends the fibers on the outside of the bend get put under tension. The fibers on the inside of the bend get put under compression. When the thickness increases the fibers have to stretch a greater distance then if the laminate is thinner. This is why increasing the thickness will increase the stiffness. BTW, a fibers resistence to stretching is called the tensile modulus. The higher the tensile modulus the more force it takes to stretch the material. Material with a higher tensile modulus, like carbon fiber, will make the structure stiffer.
The strength of the sandwich (skin/core/skin) is dependent on cores ability to hold the fibers in position. If the compression strength of the core is too low the skin under compression with buckle in toward the core and crush it. If the bond strength between the core and the skin is too low the skin under compression can buckle away from the core or the skin can shear away from the core.
Increasing the stiffness can often improve the load carrying ability of the structure. Most structures fail from buckling or deforming under compression. The stiffer the laminate is the better it will perform in a buckling situation. Buckling under compression is not the same as compression strength. I like to think of compression strength as crushing a cube. The point at which the cube is unable to rebound to it’s original dimension is it’s compression strength. Most structures will fail from buckling before the compression strength of the material is reached.