To solve the expression \( \frac{4}{5} - \left[ \left( -\frac{5}{2} \right) - \left( -\frac{3}{4} + \frac{2}{8} \right) \right] \), we need to follow the order of operations (PEMDAS/BODMAS).

1. **Simplify inside the innermost parentheses first:**
\[
-\frac{3}{4} + \frac{2}{8}
\]

Note that \(\frac{2}{8}\) simplifies to \(\frac{1}{4}\):
\[
-\frac{3}{4} + \frac{1}{4} = -\frac{2}{4} = -\frac{1}{2}
\]

2. **Substitute back into the expression:**
\[
\frac{4}{5} - \left[ \left( -\frac{5}{2} \right) - \left( -\frac{1}{2} \right) \right]
\]

3. **Simplify the expression inside the brackets:**
\[
\left( -\frac{5}{2} \right) - \left( -\frac{1}{2} \right) = -\frac{5}{2} + \frac{1}{2}
\]

Combine the fractions:
\[
-\frac{5}{2} + \frac{1}{2} = -\frac{4}{2} = -2
\]

4. **Substitute back into the expression:**
\[
\frac{4}{5} - (-2)
\]

5. **Simplify the final expression:**
\[
\frac{4}{5} + 2
\]

Convert 2 to a fraction with a common denominator of 5:
\[
2 = \frac{10}{5}
\]

Add the fractions:
\[
\frac{4}{5} + \frac{10}{5} = \frac{14}{5}
\]

So, the final result is:
\[
\frac{14}{5}
\]