  A new generalized inverse for a square matrix $H\in\mathbb{C}^{n\times n}$,
called CCE-inverse, is established by the core-EP decomposition and
Moore-Penrose inverse $H^{\dag}$. We propose some characterizations of the
CCE-inverse. Furthermore, two canonical forms of the CCE-inverse are presented.
At last, we introduce the definitions of CCE-matrices and $k$-CCE matrices, and
prove that CCE-matrices are the same as $i$-EP matrices studied by Wang and Liu
in [The weak group matrix, Aequationes Mathematicae, 93(6): 1261-1273, 2019].
