  We apply minimal weakly generating sets to study the existence of
Add$(U_R)$-covers for a uniserial module $U_R$. If $U_R$ is a uniserial right
module over a ring $R$, then $S:=$End$ (U_R)$ has at most two maximal (right,
left, two-sided) ideals: one is the set $I$ of all endomorphisms that are not
injective, and the other is the set $K $ of all endomorphisms of $U_R$ that are
not surjective. We prove that if $U_R$ is either finitely generated, or
artinian, or $I \subset K$, then the class Add$(U_R)$ is covering if and only
if it is closed under direct limit. Moreover, we study endomorphism rings of
artinian uniserial modules giving several examples.
