For Hilbert space operators $A$ and $B$, let $\delta_{AB}$ denote the generalised derivation $\delta_{AB}(X)=AX-XB$ and let $\triangle_{AB}$ denote the elementary operator $\triangle_{AB}(X)=AXB-X$. If $A$ is a $pk$-quasihyponormal operator, $A\in pk-QH$, and $B^*$ is an either $p$-hyponormal or injective dominant or injective $pk-QH$ operator (resp., $B^*$ is an either $p$-hyponormal or dominant or $pk-QH$ operator), then $\triangle_{AB}(X)=0\Longrightarrow \delta_{A^*B^*}(X)=0$ (resp., $\triangle_{AB}(X)=0\Longrightarrow \triangle_{A^*B^*}(X)=0)$.