  In this paper, we introduce the notion of weak core and central weak core
inverse in a {\it proper $*$-ring}. We further elaborate on these two classes
by producing a few representations and characterizations of the weak core and
central weak core invertible elements. We investigated additive properties and
a few explicit expressions for these two classes of inverses through other
generalized inverses. In addition, numerical examples are provided to validate
claims on weak core inverses. Following {\it proper $*$-ring} and their
interconnections with Clifford algebra, we also present examples of the group
inverse and the weak core inverse of a non-zero non-invertible quaternion
$\mathbb{H}_s$.
