A "rough patch" is a difficult time. For example, a couple could be going through a "rough patch" in their marriage.
Generally, the phrase implies a short duration, although most of the time the duration turns out to be longer. In your example, officials might say that the economy is going through a "rough patch" to try to assuage the public. The public, however, will most likely be weary of such explanations, and would likely assume the officials were not being completely truthful, thus defeating the calming intent.