Would the surface temperature. of stars classified as white dwarfs be generally higher or lower than that of stars classified as supergiants?
When a star enters the white dwarf stage of evolution it is no longer undergoing any fusion reactions, therefore it is no longer generating any energy. The temperature of the white dwarf is the residual temperature left from the nova of the star. This temperature can be very high to start (around 100,000K) but it will decrease constantly.
As long as it has a higher temperature than the background temperature of space (2-3K) it is considered a white dwarf so you could have a white dwarf at say 5 K. Once it reaches 2-3K it is called a black dwarf, although none exist or will exist for trillions of years.
The temperature of a supergiant would be dependent on the color of the star. A red super giant has a temperature of around 4000K, or much lower than our sun and about the same temperature of the coolest white dwarf known. The temperature of a blue supergiant would be somewhere around 20,000 K.
So the temperature range of a white dwarf would be from 4 K up to about 100,000 K. To cover all supergiants you would be looking at 4000K to say 50,000K. So a white dwarf could be hotter than the hottest supergiant or cooler than the coldest supergiant.
For comparison our sun is about 6,000 K.