There are a few well known constructions of potential categories of pure motives for smooth projective varieties over a field. My understanding is that modulo the standard conjectures these should be equivalent and are understood to be the correct definitions in this context.
As I understand it, the contentious part of the story begins when we consider singular varieties. In this case the known constructions of pure motives are poorly behaved and we need to think of a theory of so-called mixed motives.
An analagous situation presents itself in Gorenstein homological algebra, in which the bounded derived category of coherent sheaves on a scheme $X$ does not capture the cohomological data contained in syzygies. This problem is solved by a recollement adjoining the necessary data in acyclic complexes to the bounded derived category. For specifics see the Murfet's thesis.
Is there any hope of this approach working to define an abelian category of mixed motives?
I suppose the idea would be to define some auxiliary abelian category of which is the "motives of singularities" of $X$ and then to show that there is some abelian category which is a recollement of this new abelian category and the abelian category of pure motives.
