5. D. From $y=2 \sin \left(x+\frac{2}{3} \pi\right)$, shifting left by $m$ units gives $y=2 \sin \left(x+m+\frac{2}{3} \pi\right)$, so $m+\frac{2}{3} \pi=k \pi+\frac{\pi}{2}$, therefore $m=k \pi-\frac{\pi}{6}$. Let $k=1$, then $m=\frac{5}{6} \pi$.
5. The graph of the function $y=\sqrt{3} \cos x-\sin x$ is shifted to the left by $m$ units, and the resulting graph is symmetric about the $y$-axis. Then the smallest positive value of $m$ is ( ).
A. $\frac{\pi}{6}$
B. $\frac{\pi}{3}$
C. $\frac{2 \pi}{3}$
D. $\frac{5 \pi}{6}$