1. Write the general term

For $\left(\frac{4x}{5}+\frac{5}{2x^2}\right)^9,$ the general term is $T_{r+1}=\binom{9}{r}\left(\frac{4x}{5}\right)^{9-r}\left(\frac{5}{2x^2}\right)^r,$ where $r=0,1,2,\dots,9$.

2. Simplify the power of $x$

From the term above, $x\text{-power}=x^{9-r}\cdot x^{-2r}=x^{9-3r}.$

We need the coefficient of $x^{-6}$, so set $9-3r=-6.$ Thus, $-3r=-15\quad\Rightarrow\quad r=5.$

3. Find the required term

So the required term is $T_6$: $T_6=\binom{9}{5}\left(\frac{4x}{5}\right)^4\left(\frac{5}{2x^2}\right)^5.$

4. Compute its coefficient

First, $\binom{9}{5}=126.$

Now, $\left(\frac{4x}{5}\right)^4=\frac{4^4x^4}{5^4},$ and $\left(\frac{5}{2x^2}\right)^5=\frac{5^5}{2^5x^{10}}.$

Multiplying, $T_6=126\cdot \frac{4^4}{5^4}\cdot \frac{5^5}{2^5}\cdot x^{4-10}.$

Since $x^{4-10}=x^{-6}$, the coefficient is $126\cdot \frac{4^4\cdot 5^5}{5^4\cdot 2^5}.$

Simplify: $\frac{5^5}{5^4}=5,$ and $4^4=(2^2)^4=2^8.$ So, $\frac{4^4}{2^5}=\frac{2^8}{2^5}=2^3=8.$

Therefore, $\text{coefficient}=126\cdot 8\cdot 5=126\cdot 40=5040.$

5. Final answer

The coefficient of $x^{-6}$ is $\boxed{5040}.$