Are cloudy nights warmer than nights without clouds?
Is there a correlation between temperature and cloud, during the night?


Very much so. It's called radiative cooling. Three factors come into play: cloudiness, relative humidity, and windiness. Nighttime radiative cooling is greatest under clear skies, low relative humidity, and light or no winds. The temperature drop (in degrees per hour) can be a factor of more than four greater under conditions of clear skies, low humidity, and light winds compared to that under conditions of thick low clouds and high relative humidity. That increased cooling can make for a significant temperature drop on a long winter night.