  Let $K,\,H$ be Hilbert spaces and let $L(K,H)$ denote the set of all bounded
linear operators from $K$ to $H$. Let $A \in L(H)\triangleq L(H,H)$ with $R(A)$
closed and $X,Y \in L(K,H)$ with $R(X)\subseteq R(A),R(Y)\subseteq R(A^*)$. In
this short note, we give some new expressions of the Moore--Penrose inverse
$(A-XY^*)^+$ of $A-XY^*$ under certain suitable conditions.
