If you find some white fuzz on a zucchini, especially on the stem, we recommend that you simply cut the end off and wash it really well under running water.
When the skin on a zucchini is damaged, it excretes a sticky sap as a way of healing the skin. The sap often will mould quickly due to the sugar content of the sap, but those spots can almost always be washed off -- as the sap washes away it takes the mould growing on it with it. This sap is one of the reasons zucchini quality is always so tricky! The smallest nick will cause the sap to run! Generally, if the zucchini is nice inside when you slice it, and as long as you can wash/pare away any small nicks or mould spots, you can still use it, especially if you're cooking with it.