Upper bound for size of minimal covers of a set
The trivial bounds are that $|c| \le |X|=n$ and $|c| \le |S|=p$.

Knowing bounds on the sizes of the sets $s_i$ doesn't help much.  For instance, consider the family of sets $s_1,\dots,s_{n-k+1}$ given by $s_i=\{i,n-k+2,\dots,n-1,n\}$.  These sets are all of size $k$, and yet the largest inclusion-minimal cover has size $n-k+1$, i.e., exactly equal to the trivial bound of $|S|$ and fairly close to the trivial bound of $|X|$.  So, knowing only the sizes of the sets does not help to provide upper bounds that are much better than the trivial ones.