It depends on the form of the zinc.
The stomach will regulate its internal pH. There are lots of sensory nerve endings that react directly to the stomach’s acidity, because it is extremely important that the stomach have the right acidity. This is important both for digestion and for killing parasites.
For most any food you eat, the stomach’s acidity decreases after eating. The stomach responds immediately by closing the exit to the intestines, and by increasing its HCl excretion. This continues until acidity has reached normal levels again.
Returning to your question, if the zinc you take is in any highly acid form, the stomach will have to produce less HCl than when the zinc is in some caustic form.
Thus, it is not the zinc itself that reduces the HCl production of the stomach. It is the stomach’s normal response to the acidity of the “food” (in this case: supplements) you take.
The same applies to your HCl intake. Let’s say that normally, your stomach would have to produce 5 ml of HCl in order to maintain its acidity. Let’s assume that your supplemental HCl intake is equivalent to 2 ml of stomach-produced HCl. Then when you take your supplemental HCl, the stomach will respond by producing only 3 ml of HCl, because that will then be enough to achieve the desired acidity.