I think the following is a common point to the previous answers. It is a very general and important step in many "devissages" of mixed sheaves (perverse sheaves, D-modules, mixed Hodge modules etc...) and illustrates why resolution of singularities is so important. This is really standard but it was never clearly explained to me and I don't know a good reference for it so I thought it was worth a post. If anyone has a good reference for it I'd be glad to read it.

Let's consider a proper morphism $\pi: Y\to X$ that is a isomorphism outside $i:Z\hookrightarrow X$. Basically $\pi$ is a resolution of the singularities. Then we have a distinguished triangle
$$
F \to \pi_{*} \pi^{*} F \to i_* Q \to +1
$$
where $Q = Cone(i^* F \to i^* \pi_* \pi^* F)$ (using proper base change).

Taking cohomology (by that I mean $H^i(a_*-)$ with $a:X\to pt$) we get
$$
H^i(X,F) \to H^{i}(Y,F) \to H^i(Z,Q) \to H^{i+1}(X,F)
$$

If $\pi$ is a blowup with smooth center $Z$, we have $Q = \bigoplus_{q=1}^{c-1} i^*F(-q)[-2q]$ so this is really easy. By iterating, we get results in the case we have a nice resolution of singularities (sequence of blow-ups with smooth centers giving a normal crossing divisor). So we reduced the problem to computing cohomology on a normal crossing divisor. And by induction on the number of irreducible components this reduces to the case of a a smooth variety.

In your case what you want is to control the weights of $H^i(Z,Q)$ so the suggestion is to track the weight filtration in the previous reasoning which (in theory) isn't so hard since it is strict.

I don't know enough about alterations to comment about positive caracteristic but I think the principles are the same.