On integers of the form $p+F_{2^k}+F_q$

In 1934, Romanoff proved that the set of positive integers representable as the sum of a prime and a power of two has positive lower density. Erdős later constructed an infinite arithmetic progression of odd integers none of which admits such a representation. Let $F_n$ be the Fibonacci sequence. In this paper, we prove that the set of integers of the form $p+F_{2^k}+F_q$, where $p,q$ are primes and $k\ge 0$, has positive lower asymptotic density. The same holds for the set of integers not of this form.