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 $\lambda ⩾1$ 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 $\epsilon >0$, choose $\theta$ with $2\mathrm{/}15+\epsilon <\theta <0.99$, and set $h=X^{\theta }$. We prove that for all but $O_{\epsilon }(X\exp (-c_{\epsilon }(\log X{)}^{1\mathrm{/}4}))$ values of $x\in [X,2X]$, the short interval $(x,x+h]$ contains ${\asymp }_{\epsilon }h$ integers of the form $p+a$, where $p$ is prime and $a\in {\mathcal{A}}_{\lambda }(X)$.