A Romanoff-type theorem for a multiset of products of powers

Let $b_1,\dots,b_d\ge2$ be fixed integers, and let $k$ be their multiplicative rank. We study representations $n=a+b_1^{u_1}\cdots b_d^{u_d}$, where $a$ belongs to a set $\mathcal{A}$ of positive integers and $u_1,\dots,u_d$ are positive integers, counting distinct tuples of exponents separately. Under density and correlation assumptions on $\mathcal{A}$, we obtain a lower bound for the number of integers with many such representations, in terms of $d$ and $k$. In particular, when $\mathcal{A}$ is the set of primes or the set of positive integers representable as a sum of two squares, a positive proportion of the integers $n\le x$ have at least $c_1(\log x)^{d-1}$ or $c_1(\log x)^{d-1/2}$ representations, respectively, for some $c_1>0$ and all sufficiently large $x$. No multiplicative independence or coprimality assumptions on the bases are required.