Let $P$ be the point $(1,0)$ and $Q$ a point on the locus $y^2=8x$. The locus of midpoint of $PQ$ is:

Let $Q$ be $(h,k)$. Since $Q$ lies on $y^2=8x$, we have $k^2=8h$.Let $(x,y)$ be the midpoint of $PQ$, where $P=(1,0)$.Using the midpoint formula:
$$x=\frac{h+1}{2}⟹h=2x-1$$
$$y=\frac{k+0}{2}⟹k=2y$$
Substitute $h$ and $k$ in the equation $k^2=8h$:
$$(2y{)}^2=8(2x-1)$$
$$4y^2=16x-8$$
Dividing by 4:
$$y^2=4x-2⟹y^2-4x+2=0$$