Let $X/K$ be a variety (scheme of finite type, geometricaly integral) over a finitely generated field $K$. If it is smooth and proper, we can formulate the Tate conjecture, and if $\text{char}(K) = 0$ also the Hodge conjecture.

Are there analogues in the general case (when cohomology is no longer pure)?

Of course there is a conjectural generalisation that says:

The $\ell$-adic (resp. Betti) realisation functors from the category of mixed motives over $K$ to the category of Galois representations (resp. mixed Hodge structures) is fully faithful.

*(NB: "category of Galois representations" means that morphisms are equivariant for some open subgroup of $\mathrm{Gal}(\bar{K}/K)$.)*

The problem with this generalisation is that the statement refers to the category of mixed motives, and we don't know whether it exists (i.e., the conjecture is conjectural).

Q:Is there a generalisation that does not depend on the category of mixed motives (probably involving some sort of cycle maps)?

On the one hand I expect this is possible, on the other hand maybe the same problem that pops up in defining mixed motives also appears when trying to formulate a generalisation of the Tate and Hodge conjectures. (I am still very much a novice when it comes to the mixed versions of motives and cohomology.)

Note that there is a good generalisation of the Mumford–Tate conjecture. After all, we still have Mumford–Tate groups and images of Galois; we just don't expect them to be reductive (and in general they are not).

For some reason I could not find anything about this via a search on the general internet or specifically mathscinet.