I just experimented with some small primes and obtained $7=2^2+3$ and $13=2^3+5$ and $17=2^2+13$ and $23=2^4+7$.
All these primes are sum of a prime and a power of $2$.
So it naturally led me to conjecture:
There is an infinite number of primes $p$ so that $p=2^k+q$, where $q$ is another prime and $k \in \mathbb N$
You can see that it is also allowed that $k=1$ so, if there were an infinite number of twin primes (or any pairs of primes that differ by some fixed $2^k$) then this conjecture would be trivially true.
But, this is far weaker conjecture because it allows all powers of $2$ at once.
Is this known?
There is a conjecture that for every even positive integer $d$, there are infinite many pairs of primes $p,q$ with $p+d=q$. This implies this conjecture. In fact, for every power of $2$, we would only need one such pair.
Another heuristic for the conjecture. For every prime power $2^k$, there are infinite many odd numbers $o$, such that $2^k+o$ is prime. It seems very likely that there is at least one prime number (probably even infinite many prime numbers) among those odd numbers.
Nevertheless, I am sceptical about a proof of this conjecture, although at first glance, it does not appear to be too difficult.