The main references for this question are
2 : the book "Lecture notes in motivic cohomology" written by Carlo Mazza, Vladimir Voevodsky and Charles Weibel.
Context: in both 1 and 2 we can find a possible definition of motivic cohomology using $Hom$ in the category of geometric motives over $k$. I begin to recall the main ideas of this.
This category looks to be in 1 the pseudo-abelianisation of a localisation of the bounded complexes of smooth schemes over a field $k$, and it is in 2 the same construction, but now on the category of sheaves (of $R$-algebra) with transfers for the Nisniech site (questions: why this particular site? Do we obtain the same category if $R=\mathbb{Z}$?). Then the category of effective Chow motives over $k$ embeds through a functor $M$ into effective geometric motives over $k$ (see proposition 20.1 of 2). Now, given a smooth scheme $X$ we define as in [2, definition 14.16] the motivic cohomology (for a ring $R$) to be $$H^{n,i}_{\text{mot}}(X,R) \overset{\text{def}}{=}Hom_{\text{geo. motives}}(M(X),R(i)[n]).$$
My question is: Are we able to generalize the above construction, replacing $R$ by a sheaf. In particular, I intersting in the case where $R=M$ whith $M$ a Chow motive over $X$ seen as a locally constant sheaf over $X$. If the answer is "Yes and it is exactly the same construction" then is there any reference which deals with? More generally do you have any reference for motivic cohomology of a scheme $X$ with coefficient in a Chow motive over $X$?