Given, the line with slope $m$ and $y$ intercept $4$.

As we know by the slope-intercept form of a line, a line with a slope $m$ and $y$ intercept $c$, the equation of the line is $y=mx+c$.

$\Rightarrow$The equation of the line is $y=mx+4$.

Also, the distance between a point $\left(x_1,y_1\right)$ and a line $ax+by+c=0$ is given by $\left|\frac{a{x}_1+b{y}_1+c}{\sqrt{a^2+b^2}}\right|$ units.

$\therefore$ The distance between origin and the line $y=mx+4$ $=\frac{4}{\sqrt{1+m^2}}$.