The simple answer: It will! But not as much as you would think!
Photovoltaic (PV) cells will convert any kind of sunlight—direct, diffuse, and reflected—into energy. While it’s true that PV cells have the highest output with direct sunlight, that doesn’t mean they simply stop providing value when the clouds come rolling through. Even at a fraction of peak production, a solar installation is still adding value to an energy portfolio. It's important to keep in mind that clouds come and go but a solar installation’s value builds year after year. Neither the peaks nor the valleys are representative of the average added value.
Cooler weather often associated with cloudy weather can actually improve the efficiency of a solar array. In nearly all technology, heated circuits operate far less efficiently than cooled circuits, and the same applies to PV cells. While a solar array in a hot, arid location may receive more direct sunlight, an array in a more temperate location—even with higher average cloud cover—might outperform the arid installation simply because what sunlight it gets is converted more efficiently. On top of that, an occasional rain effectively washes solar panels and helps prevent soiling which would otherwise drastically impact production. In fact, in arid zones there are entire micro-industries devoted to simply cleaning solar panels because of the lack of rain.