Short intervals for the Romanoff-type sumset

Let $X$ be large and let $\mathcal{P}$ denote the set of primes. Fix positive real parameters $r_1,\dots,r_s$ and a parameter $\lambda\geqslant 1$ determined by a balancing relation, and let $\mathcal{A}_{\lambda}(X)\subset[1,2X]$ be the associated lacunary set generated by sums of powers of $2$ with polynomially growing exponents. Set $\mathcal{S}_{\lambda}:=\mathcal{P}+\mathcal{A}_{\lambda}(X)$. Fix $\varepsilon>0$, choose $\theta$ with $2/15+\varepsilon<\theta<0.99$, and set $h=X^{\theta}$. We prove that for all but $O_{\varepsilon}\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)$ values of $x\in[X,2X]$, the short interval $(x,x+h]$ contains $\asymp_{\varepsilon} h$ integers of the form $p+a$, where $p$ is prime and $a\in\mathcal{A}_{\lambda}(X)$.