Less dew is formed on cloudy nights because of

in cloudy nights the radiation takes place very slowly

In cloudy nights the radiation takes place very quickly

clouds scatter moisture

clouds absorbs the falling dew

Radiation is very slow on cloudy nights, resulting in less condensation on cloudy nights.

On cloudy nights, the clouds are thick and trap heat and water vapor in the clouds, resulting in very slow radiation.

This all shows why dew levels drop on cloudy nights.