  The notion of the weighted core inverse in a ring with involution was
introduced, recently [Mosic et al. Comm. Algebra, 2018; 46(6); 2332-2345]. In
this paper, we explore new representation and characterization of the weighted
core inverse of sum and difference of two weighted core invertible elements in
an unital ring with involution under different conditions. Further, we discuss
reverse order laws and mixed-type reverse order laws for the weighted core
invertible elements in a ring.
