Can two neighbouring primes average to a prime

Take two primes $p<q$ with no prime strictly between them. Is it possible for $\tfrac{p+q}{2}$ to be prime?

No. For $p=2,q=3$ the mean $2.5$ is not an integer. Otherwise the mean is an integer lying strictly between $p$ and $q$, and since $p,q$ are consecutive primes, no integer in that gap is prime.