To add these fractions, we first need a common denominator. The least common multiple of 3 and 4 is 12, so we need to find equivalent fractions with denominators of 12. We have \[\begin{aligned} \dfrac{8}{3} &= \dfrac{8 \cdot 4}{3 \cdot 4} = \dfrac{32}{12},\text{ and} \\ \left(-\dfrac{9}4\right) &= \left(\dfrac{-9}4\right)\cdot \dfrac{3}{3} = \dfrac{-27}{12}. \end{aligned}\] The sum of these fractions is \[\dfrac{32}{12}+\dfrac{-27}{12}=\dfrac{32+(-27)}{12}=\dfrac{5}{12}.\] So $\dfrac{8}{3}+\left(-\dfrac{9}4\right) = \boxed{\dfrac{5}{12}}.$
