Can you have a low pressure with no clouds?
Low pressure areas are simply areas where the air is less dense than the surrounding areas.  The scale of these differences can be small, size of a small farm field, or large, covering expansive areas of the earth's surface.  

Clouds form around low pressure systems typically because of the way low pressure systems develop and the air moves around them.  Typically low pressure systems start forming over relatively warmer sections of the earth, this is gross over simplification, as the air warms it becomes less dense and will tend to rise.  This rising of warm often moist air leads to cloud formation because as air rises it also becomes cooler, see Lapse Rate.  Typically at some time the temperature drops below the dew point in the air moving upwards and clouds form.  But if the relative humidity of the air is not high it is possible that the air mass in the low pressure system never rises high enough for condensation of the water vapor to occur, in this case no clouds will form.

In conclusion it is possible to read a lower air pressure at a site and still not have cloud formation.

This site has a good graphic on how low and high pressure systems relate to each other. High and Low weather systems described

Please note that I am not meteorologist.  This answer does not consider all the aspects associated with cloud formation such as how airborne particulates can affect condensation