Removing mold on wall beneath sink

DA01's theory is correct. Outside humidity is migrating through the wall until it contacts material colder than the dew point of the vapor laden air. The vapor at that point condenses into liquid water. Aside from not using A/C, one remedy is to install a vapor barrier between the insulation and the exterior wall cinder block. This has a detrimental effect in the winter because the situation reverses. You need a vapor barrier on the interior side of the insulation. This is feasible, but the problem now is any moisture that gets past the barrier (no barrier is perfect) has no where to go. It is trapped in the insulation.

This of course means removing all insulation and drywall on exterior walls. Thus, I believe the best course of action is to insulate with a material unaffected by moisture and is actually a vapor barrier itself. One such insulation is spray applied polyurethane. The only drawback of this material is that it is quite expensive. It's the ideal material for difficult, varying climatic conditions such as yours.

It's hard to tell from the picture but it looks like it was tile mounted to a wood board. Wood is one of the worst materials for behind the sink. If going to replace with tile, use a concrete backer board instead and consider using tile trim caps to cover the top. Otherwise use a solid stone or stone like material.

And make sure the caulking is in good shape. Every couple of years it needs to be stripped out and replace. If it is starting to look old and cracked, it's been there too long. No mater what the material, water under or behind the splash with always cause problem.s