"Redundant" finite subcovering of a compact space.

First, take a finite cover of your space $M.$

Note that since it is finite, the set of point $p \in M$ such that $p$ is only in one of your $\mathcal{U_i}$ will be a closed set. (You need to use the finiteness of your cover here)

Since this is closed, it will be compact (here I am assuming that your space is Hausdorff though). Then consider a cover of that set by elements of $\mathcal{U}$ that are not in your first cover.