Stephen Wilde, who has been a member of the Royal Meteorological Society since 1968, answers the question:
The answer depends on whether it is day or night.
Note that the question is only valid for individual clouds because clouds that form part of horizontally advecting air masses involve different thermal mechanisms.
The question is also only valid for stratiform or fair weather cumulus clouds which are not actively involved in upward convection because surfaces beneath active convective cells are affected by winds flowing in from around the convecting cell.
In daytime an individual cloud both reduces convection and increases humidity beneath it which allows indirect, even diffuse, insolation to continue warming the surface and air beneath the cloud. That is why a cloud passing over a water surface can cause a rise in temperature beneath it. The condensate in the cloud blocks cooling convection and the higher humidity reduces cooling evaporation from the water surface. AGW theory says that the cloud radiates IR downward to raise the surface temperature which I consider to be wrong.
At night time an individual cloud reduces radiative loss to space because the cloud is at (or sometimes above, due to advection) the temperature along the lapse rate slope which is attributable to its height. For maximum radiative loss one needs an atmosphere that is perfectly transparent to IR and the presence of any material reducing that transparency must reduce the rate of radiative loss.
An interesting factor at this point is that IR is susceptible to transparency being reduced by the pressure and density of non-radiative gases because collisional activity can divert IR from photon release to conduction and convection. That is less of an issue with more energetic wavelengths which are not so affected by pressure and mass density. One of the mistakes of AGW radiative theory is to fail to see that IR behaves differently to other wavelengths in the presence of mass compressed by gravity. See the Catling paper in support of that point.
The night time cloud cannot warm the ground but it can reduce radiative cooling which then becomes limited to the temperature of the lapse rate slope at cloud base.
The ground will cool to a lower temperature beneath higher clouds because they are at a higher and colder location along the lapse rate slope.
Clouds at the surface (fog) can stop surface radiative cooling altogether if the fog is deep enough but sometimes the fog layer is shallow and allows radiative cooling of the surface to continue until freezing fog forms.
Clouds high up, for example cirrus clouds, may inhibit radiative cooling of the surface very little.
Note though that the presence of clouds at night is generally indicative of advection of warmer air across the radiating surface so that the amount of radiative inhibition caused solely by the presence of the cloud is hard to separate out.
A surface rarely cools to the temperature of a high cloud because other factors are involved in the cooling process such as the length of day and night, the energy stored in the surface materials, the amount of local air mixing and the regional synoptic situation.
In no case is downward IR from the cloud doing any active warming. In all cases the cloud is simply reducing the capability of the surface to cool itself using different mechanisms day and night.