Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of $2$?

Gallagher [Ga75] has shown that for any $ϵ > 0$ there exists $k(ϵ)$ such that the set of integers which are the sum of a prime and at most $k(ϵ)$ many powers of $2$ has lower density at least $1 - ϵ$.

Ref: Gallagher, P. X., _Primes and powers of 2_.