I've been thinking for some months about a slightly weaker form of Goldbach's conjecture: namely that every large enough integer is the sum of two prime powers or primes, that is $\exists C>0,\forall n>C,n=p^a+q^b$ with $p$ and $q$ primes and $a$ and $b$ positive integers. Of course if $n$ is odd, the truth of such a statement requires exactly one the two primes $p$ or $q$ to equal $2$.
I would like to know if the hypothetical exceptional set $E(x):=\#\{n\le x\mid n\equiv 1\pmod 2,\forall 0<k<\frac{\log n}{\log 2},\Lambda(n-2^k)=0\}$, where $\Lambda$ is the von Mangoldt function, is an $O(x^{1-\eta})$ for some $\eta>0$, conditionally or unconditionally.