Summary
I wonder if there is a solution to a simpler (or at least less strict) version of the Set Cover Problem where instead of searching for the smallest subset of $S$ whose union is equal $U$, I instead seek the smallest subset of $S$ whose union is equal to or contains a specific subset of $U$.
Example
Let the universe, $U$, be $$U = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \}$$ and the set $S$ (whose union equals $U$) be $$S = \{ \{ 1, 2, 3 \}, \{ 2, 6 \}, \{ 4, 5, 7 \}, \{ 8, 9, 10, 3 \}, \{ 9, 10 \} \}$$
Is there an algorithm to find the smallest subset $X$ of $S$ whose union equals (or contains all of), say, $B$ $$B = \{ 1, 2, 3, 4, 5 \}$$ (or any other specific subset of $U$) that is better than brute forcing all combinations and checking? $X$ must either contain all of or be equal to B.
(I'm not looking for the solution to this specific example, but an algorithm for the general case)
There can't be any difference in the complexity of the "simpler" version, because it's clearly equivalent.
As above, say $S$ is a collection of subsets of $U$ with union $U$, and say $B\subset U$. Define $$S_B=\{E\cap B:E\in S\}.$$Then finding a subset of $S$ with union containing $B$ is the same as finding a subset of $S_B$ with union equal to $B$; this last is precisely an instance of the original SCP.