For a smooth projective variety $X$ and its closed nonsmooth subvariety $Z$ I would like to say that a cone of the morphism between the motivic cohomology of $Z$ and those of $X$ is the motivic cohomology of $X\setminus Z$ with compact support. Is this statement known over a positive characteristic (perfect) base field (at least, when $Z$ is a smooth normal crossing divisor; I am interested in $l$adic motivic cohomology)? Also, I would like here to relate the motivic cohomology of $Z$ with those of its Henselization in $X$; cf. On (the cohomology of) Hensel pairs
