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 $\lambda>0$ 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 $2/15+\varepsilon<\theta<1-\delta_0$, where $\delta_0>0$ is an absolute constant, 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 between $c_\varepsilon h$ and $C_\varepsilon h$ integers of the form $p+a$ with $p\in\mathcal P$ and $a\in\mathcal A_\lambda(X)$.