To evaluate the expression
\[ \dfrac{9}{m} + 4 \]
when \( m = 3 \), we first substitute \( m \) with 3:
\[ \dfrac{9}{3} + 4 \]
Next, we calculate \( \dfrac{9}{3} \):
\[ \dfrac{9}{3} = 3 \]
Now we add 4 to this result:
\[ 3 + 4 = 7 \]
Therefore, the final result is
\[ \boxed{7} \]