Given,

The point $\left(a, 8,-2\right)$ divides the line segment joining the points $\left(1,4,6\right)$ and $\left(5,2,10\right)$ in the ratio $m:n$,

So, by section formula we get,

$a=\frac{5m+n}{m+n} ......\left(1\right)$

$8=\frac{2m+4n}{m+n} .......\left(2\right)$

$-2=\frac{10m+6n}{m+n} ..........\left(3\right)$

Now solve either equation $\left(2\right) \text{or }\left(3\right)$ we get, $\frac{m}{n}=\frac{-2}{3}$

Now putting in equation $\left(1\right)$ we get $a=-7$,

Now putting the values in $\frac{2m}{n}-\frac{a}{3}=\frac{-2\times 2}{3}-\frac{-7}{3}=\frac{{\displaystyle 7-4}}{3}=1$.