To evaluate the expression \(\dfrac{9}{m} + 4\) when \(m = 3\), we can substitute \(m\) with \(3\) in the expression.
First, we substitute:
\[ \dfrac{9}{3} + 4 \]
Next, we calculate \(\dfrac{9}{3}\):
\[ \dfrac{9}{3} = 3 \]
Now we add \(4\) to \(3\):
\[ 3 + 4 = 7 \]
Thus, the value of \(\dfrac{9}{m} + 4\) when \(m = 3\) is
\[ \boxed{7}. \]