Using the Cech spectral sequence, we can compute the cohomology of a topological space via the cohomology of open subsets. Is there a version of this where we use cohomology with compact support? Let me describe what I'd like: If $p:X\to \text{Spec }k$ is a scheme (or a topological space), and $(U_i)_i$ a finite open covering, do we have $$(*) \quad E_2^{p,q}=\check{\mathrm{H}}^n_c(\mathfrak{U},\mathcal{F})\Rightarrow H^n_c(X,\mathcal{F})?$$ Since $H^n_c(X,\mathcal{F})=R^ip_{!} \mathcal{F}$, if we can show that the composition of the forgetful functor $F:\text{Sh}(X)\to \text{PSh}(X)$ with $\check{\mathrm{H}}^0_c(\mathfrak{U},-):\text{PSh}(X)\to \text{Sh}(\text{Spec }k)$, where $$\check{\mathrm{H}}^0_c(\mathfrak{U},F)(\text{Spec }k)=\text{ker}\left(\prod_i (p\vert_{U_i})_!\mathcal{F}(k) \to \prod_{i,j} (p\vert_{U_i\cap U_j})_!\mathcal{F}(k) \right),$$ agrees with $p_!$ (i.e. $p_!=\check{\mathrm{H}}^0_c(\mathfrak{U},-)\circ F$), then $(*)$ would just be the consequence of the Grothendieck spectral sequence. Is that true, and is there a nice reference for this?
Edit: Maybe I ought to add that there exist one case where this is true: if $X=U\cup V$, then the spectral sequence degenerates at $E_2$ and yield Mayer-Vietoris, which is true for cohomology with compact support.