High pressure is normally associated with a clear sky and light winds. If a high pressure system is over a region the weather is normally quite fair! But, that is not always the case. A “Dirty” high is a high pressure system that has some clouds or light precipitation.
High pressure causes air to sink making it very difficult for mid to high-level clouds to form. Sinking air warms and dries. While clouds form when air rises, cools, and condenses.
If the high pressure is associated with cool air but moves over a saturated ground this can leads to low-level clouds forming. The moisture on the ground evaporates into the low levels of the atmosphere and leads to clouds or fog to form. High pressure has light winds and a clear sky associated with it and this can lead to fog forming overnight and into the morning hours. A temperature inversion can form with high pressure and that could trap moisture near the ground and allow clouds and fog to form.
Also if there are lifting mechanism such as orographic lift (the forcing of air to rise due to the terrain i.e mountains), Warm Air Advection (warmer air moving into a cooler place due to wind), or an Upper-Level Divergence (when an air mass is pulled away from a region faster than that air mass can be replaced) can lead to some clouds to form.
A strong high pressure is less likely to have clouds form. If a high pressure has a pressure of 1025 millibars or more it is considered a strong high pressure.