Andrew has $10$ balls in a bag, each a different color. He randomly picks a ball from the bag $4$ times, with replacement. The expected number of distinct colors among the balls he picks is $\tfrac{p}{q}$, where $\gcd(p, q) = 1$ and $p, q > 0$. What is $p + q$?

1. **Define the problem and the variables:**
   - Andrew has 10 balls, each of a different color.
   - He picks a ball 4 times with replacement.
   - We need to find the expected number of distinct colors among the balls he picks.

2. **Calculate the probability that a specific color is not picked in one draw:**
   - The probability that a specific color is not picked in one draw is \(\frac{9}{10}\).

3. **Calculate the probability that a specific color is not picked in 4 draws:**
   - Since the draws are independent, the probability that a specific color is not picked in 4 draws is \(\left(\frac{9}{10}\right)^4\).

4. **Calculate the probability that a specific color is picked at least once in 4 draws:**
   - The probability that a specific color is picked at least once in 4 draws is \(1 - \left(\frac{9}{10}\right)^4\).

5. **Calculate the expected contribution of one color to the total number of distinct colors:**
   - The expected contribution of one color to the total number of distinct colors is \(1 - \left(\frac{9}{10}\right)^4\).

6. **Use the linearity of expectation to find the total expected number of distinct colors:**
   - By the linearity of expectation, the expected number of distinct colors is the sum of the expected contributions of each color.
   - Since there are 10 colors, the expected number of distinct colors is \(10 \times \left(1 - \left(\frac{9}{10}\right)^4\right)\).

7. **Simplify the expression:**
   \[
   10 \times \left(1 - \left(\frac{9}{10}\right)^4\right) = 10 \times \left(1 - \frac{6561}{10000}\right) = 10 \times \frac{3439}{10000} = \frac{34390}{10000} = \frac{3439}{1000}
   \]

8. **Identify \(p\) and \(q\) and ensure they are coprime:**
   - Here, \(p = 3439\) and \(q = 1000\).
   - The greatest common divisor (gcd) of 3439 and 1000 is 1, so they are coprime.

9. **Calculate \(p + q\):**
   \[
   p + q = 3439 + 1000 = 4439
   \]

The final answer is \(\boxed{4439}\).