More sum-product type counterexamples: products with shifts and $AA+A$

Adapting the construction disproving the sum-product conjecture over $\mathbb{R}$ present in Bloom, Sawin, Schildkraut and Zhelezov, we show the existence of a constant $c>0$ and arbitrarily large finite sets $A\subseteq \mathbb{R}$ such that $$\mathrm{\mid }AA+A+A\mathrm{\mid }\ll \mathrm{\mid }A{\mathrm{\mid }}^{2-c}\mathrm{.}$$ As a corollary, all of the sets $A+A$, $AA$, $(A+1)(A+1)$, $A(A+1)$ and $AA+A$ are of size $O(\mathrm{\mid }A{\mathrm{\mid }}^{2-c})$ for this construction.