Will a white object be cooler than a mirror when exposed to direct sunlight? Or will it be the other way around? If mirrors bounce all visible light and some of the other elements of electromagnetic spectrum, my guess is that it will be cooler than white objects–which only scatters light and not bounce them off. Do they reflect most infrared? What about UV light–which are absorbed by black objects?
The answer to this question depends on various aspects. For instance when you say white object, do you mean perfectly white? or white with respect to only visible light? Same happens for a mirror too.
The most direct way to look into the problem would be following. Mirrors are nothing but extremely fine and optically flat white surfaces at the back of glass. Usually its silver coating. Silver when left alone as a lump can act as a white object too. Now we can reformulate the question by asking: Will a fine surface of silver absorb less heat than a lump of silver? The answer is their absorption co-efficient is same. So heat absorbed per unit mass will be same.
But mirror looks more brighter as if it reflects more light than white object. The answer to this is that usually white objects do not have flat surface. So the reflection is not regular unlike mirror.
P.S. If you consider the glass of mirror through which light has to pass, you'll find that some light is absorbed their. So reflectivity of mirror is actually less.