As far as I understand, for a smooth variety $X$ its motivic cohomology could be described as the corresponding piece of the $\gamma$-filtration of (Quillen's) $K^*(X)$; this is completely true for $\mathbb{Q}$-coefficients, and true up to bounded denominators for $\mathbb{Z}$-coefficients.

My question is: is there a similar result for singular varieties? Here for motivic cohomology I would like to take $Hom_{DM}(M(X),\mathbb{Z}(p)[q]$; $DM$ is the category of Voevodsky's motives, and $M(X)$ is the motif of $X$ (I don't want to take the motif with compact support instead). These cohomology theories satisfy cdh-descent, but have no easy descriptions in terms of complexes of algebraic cycles. Unfortunately, I don't know much about the $\gamma$-filtration of $K$-theory.

Actually, I would like to prove the following fact: if a morphism $X\to Y$ of varieties induces an isomorphism for $K^*$, then the exponents of the kernels and cokernels of the corresponding morphisms for motivic cohomology are bounded (by a constant that depends only only on the dimensions of $X$ and $Y$). Any hints and/or references would be very welcome!