An improved lower bound for odd integers not of the form $p+2^a+2^b$

Let $x$ be sufficiently large and \[ N(x)=\big|\bigl\{n\le x:n\ \text{is odd and }n\ne p+2^a+2^b \textrm{ with } p \text{ a prime and } a,b\in \mathbb{N}\bigr\}\big|. \] Motivated by Crocker's result \[ N(x)\gg \log\log x, \] Erd\H os repeatedly asked whether there is an absolute constant $c_0$ such that $N(x)>c_0x$. Pan \cite{Pan} proved in 2011 that \[ N(x)\gg x\exp\!\left(
-C_0\frac{\log\log\log\log x}{\log\log\log x}\log x \right), \] where $C_0>0$ is an absolute constant. We improve on Pan's result by showing that, given any $\eta>0$, for all sufficiently large $x$, \[
N(x)\gg_\eta x\exp\left(-(4+\eta)\frac{\log\log\log x}{\log\log x}\log x\right). \]