To solve the given mathematical expression, we need to follow the order of operations (PEMDAS/BODMAS):

\[ g) \frac{5}{9} \cdot 0.2 + \left( \frac{1}{5} \right)^3 \div \left( -\frac{3}{10} \right)^3 - 2020^0 \]

1. **Exponentiation:**
\[ 2020^0 = 1 \]
\[ \left( \frac{1}{5} \right)^3 = \frac{1}{125} \]
\[ \left( -\frac{3}{10} \right)^3 = -\frac{27}{1000} \]

2. **Multiplication and Division:**
\[ \frac{5}{9} \cdot 0.2 = \frac{5}{9} \cdot \frac{2}{10} = \frac{5 \cdot 2}{9 \cdot 10} = \frac{10}{90} = \frac{1}{9} \]
\[ \frac{1}{125} \div -\frac{27}{1000} = \frac{1}{125} \cdot -\frac{1000}{27} = -\frac{1000}{125 \cdot 27} = -\frac{1000}{3375} = -\frac{4}{27} \]

3. **Addition and Subtraction:**
\[ \frac{1}{9} + \left( -\frac{4}{27} \right) - 1 \]

To combine the fractions, we need a common denominator. The common denominator for 9 and 27 is 27:
\[ \frac{1}{9} = \frac{3}{27} \]
\[ \frac{3}{27} - \frac{4}{27} = -\frac{1}{27} \]

Now, subtract 1:
\[ -\frac{1}{27} - 1 = -\frac{1}{27} - \frac{27}{27} = -\frac{28}{27} \]

So, the final result is:
\[ g) -\frac{28}{27} \]