Is every composite number the average of two primes? I'm interested in this question because it relates to a bad joke about people in their prime.

It seems to work for the first 20 numbers:

4 is the average of 5 and 3.
6 is the average of 5 and 7.
8 is the average of 5 and 11.
$\vdots$
16 is the average of 13 and 19.
$\vdots$

It would not be hard to write a program that checks more cases, but I suspect the hypothesis is true. If that is the case, then a more mathematical approach is needed.
So you are asking whether for every composite number $n$ there exist primes $p,q$ such that
$$n=\frac{p+q}{2}$$
That is, $2n=p+q$, so you are asking whether $2n$ can be written as the sum of two primes.

The question whether every even integer greater than $2$ is a sum of two primes is a famous open problem known as the Goldbach conjecture.
I think that it is all about prime gap.

If all natural numbers larger than 4 can be proved that it is the average of primes, the Goldbach's Conjecture will be solved, immediately