To solve the expression \(\frac{4}{3} + \left( \frac{1}{1 + \frac{5}{6}} \right) - \frac{5}{4}\), we will evaluate each part step-by-step.

1. First, simplify the inner expression \(\frac{1}{1 + \frac{5}{6}}\):
   • The expression inside the parentheses is \(1 + \frac{5}{6}\).
   • Convert \(1\) to a fraction with the same denominator as \(\frac{5}{6}\), which is \(\frac{6}{6}\).
   • Now, add the fractions: \(\frac{6}{6} + \frac{5}{6} = \frac{11}{6}\).
   • Now, we take the reciprocal of \(\frac{11}{6}\), which gives us \(\frac{6}{11}\).
2. Substitute \(\frac{6}{11}\) back into the original expression, so it becomes \(\frac{4}{3} + \frac{6}{11} - \frac{5}{4}\).
3. Find a common denominator to add and subtract these fractions. The denominators are \(3\), \(11\), and \(4\).
   • The least common multiple (LCM) of \(3\), \(11\), and \(4\) is \(132\).
4. Convert each fraction to have a denominator of \(132\):
   • Convert \(\frac{4}{3}\): \(\frac{4}{3} = \frac{4 \times 44}{3 \times 44} = \frac{176}{132}\).
   • Convert \(\frac{6}{11}\): \(\frac{6}{11} = \frac{6 \times 12}{11 \times 12} = \frac{72}{132}\).
   • Convert \(\frac{5}{4}\): \(\frac{5}{4} = \frac{5 \times 33}{4 \times 33} = \frac{165}{132}\).
5. Combine the fractions:
   • Add and subtract the numerators: \(176 + 72 - 165 = 83\).
   • Place the result over the common denominator: \(\frac{83}{132}\).

Thus, the value of the expression is \(\frac{83}{132}\), which matches the given correct option.