Is $k+p$ prime infinitely many times?

This is a consequence of Polignac's conjecture. It also implies the twin prime conjecture. Therefore your conjecture is intermediate in strength between those two.

There are some results known. It has been proven that there is at least one $k \leq 246$ that appears infinitely often as a prime gap.

Furthermore, assuming respectively the Elliott–Halberstam conjecture and its generalisation, one can prove that there is at respectively at least one $k \leq 12$ and $k \leq 6$ that appears infinitely often as a prime gap.

Of course, a prime gap is stronger than your condition, since the primes don't have to be consecutive.