To solve the given mathematical expressions, let's break them down step by step.

### Part b)
\[ \left(2 \frac{1}{5} - 1 \frac{2}{3}\right)^3 \]

First, convert the mixed numbers to improper fractions:
\[ 2 \frac{1}{5} = \frac{11}{5} \]
\[ 1 \frac{2}{3} = \frac{5}{3} \]

Now, perform the subtraction:
\[ \frac{11}{5} - \frac{5}{3} \]

To subtract these fractions, find a common denominator, which is 15:
\[ \frac{11}{5} = \frac{33}{15} \]
\[ \frac{5}{3} = \frac{25}{15} \]

Subtract the fractions:
\[ \frac{33}{15} - \frac{25}{15} = \frac{8}{15} \]

Now, raise the result to the power of 3:
\[ \left(\frac{8}{15}\right)^3 = \frac{8^3}{15^3} = \frac{512}{3375} \]

### Part c)
\[ 3^2 + 4 \cdot \left(\frac{7}{9}\right)^0 + \left[(-5)^2 \cdot \frac{1}{5}\right] : 25 \]

First, evaluate each part separately:
\[ 3^2 = 9 \]

Since any number raised to the power of 0 is 1:
\[ \left(\frac{7}{9}\right)^0 = 1 \]

So:
\[ 4 \cdot 1 = 4 \]

Next, evaluate the expression inside the brackets:
\[ (-5)^2 = 25 \]
\[ 25 \cdot \frac{1}{5} = 5 \]

Now, divide by 25:
\[ 5 : 25 = \frac{5}{25} = \frac{1}{5} \]

Finally, add all the parts together:
\[ 9 + 4 + \frac{1}{5} \]

Convert 9 and 4 to fractions with a common denominator of 5:
\[ 9 = \frac{45}{5} \]
\[ 4 = \frac{20}{5} \]

Add the fractions:
\[ \frac{45}{5} + \frac{20}{5} + \frac{1}{5} = \frac{66}{5} \]

So, the final results are:
- Part b) \(\frac{512}{3375}\)
- Part c) \(\frac{66}{5}\)