What number should be subtracted from

Let the required number be x.

Now,

$\frac{-2}{3}-x=\frac{-1}{6}$

$\Rightarrow \frac{-2}{3}-x+x=\frac{-1}{6}+x \quad$ (Adding $x$ to both the sides)

$\Rightarrow \frac{-2}{3}=\frac{-1}{6}+x$

$\Rightarrow \frac{-2}{3}+\frac{1}{6}=\frac{-1}{6}+x+\frac{1}{6} \quad$ (Adding $\frac{1}{6}$ to both the sides)

$\Rightarrow\left(\frac{-4}{6}+\frac{1}{6}\right)=x$

$\Rightarrow\left(\frac{-4+1}{6}\right)=x$

$\Rightarrow \frac{-3}{6}=x$

$\Rightarrow \frac{-1 \times 3}{2 \times 3}=x$

$\Rightarrow \frac{-1}{2}=x$

Hence, the required number is $\frac{-1}{2}$.