As suggested by Denis Nardin I am moving my comment here. However I don't know details well enough, so I am making this cw in case somebody can fill them in.
So, choose a cohomology theory $h^*$ like e. g. $K$-theory, and, given a cover $(U_i)_{i\in I}$ of $X$, let $p:Y\to X$ be the canonical map $\coprod_{i\in I}U_i\to X$. Then consider the associated simplicial space $$ \check C(p):=\left(Y\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix} Y\times_XY\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix}Y\times_XY\times_XY\begin{smallmatrix}\leftarrow\\\vdots\\\leftarrow\end{smallmatrix}\cdots\right), $$ so that $\check C(p)_n=\coprod_{i_1,...,i_n}U_{i_1}\times_X\cdots\times_XU_{i_n}$.
Under certain (I believe mild) conditions the induced map from its geometric realization to $X$ induces isomorphism on $h^*$. On the other hand - see e. g. Theorem 5.83 on page 163 of "Generalized Cohomology" by Kōno and Tamaki - there is a spectral sequence converging (again under some mild conditions) to $h^*$ of the geometric realization, with the second page given by cohomology of the cochain complex corresponding to the cosimplicial abelian group $h^*(\check C(p))$.
In our case - yet again under some conditions - the Moore normalization of the latter complex (given by restricting to nondegenerate simplices of $\check C(p)$) has $\prod_{\{i_1,...,i_n\}\subseteq I}h^*(U_{i_1}\cap\cdots\cap U_{i_n})$ in the $n$th degree, with understandable differentials.
