Integer parts of real powers in two Erd\H{o}s problems of Romanoff type

We study two additive problems which go back to Erdős' work around Romanoff's theorem and in which the sparse summand is a sequence of powers. In the prime case we prove a quantitative metric form of the Erdős--Kalmár problem: if $$
\mathcal{S}_y=\{p+\lfloor y^k\ rfloor:p\in\mathcal{P},\ k\ge1\},
\qquad
\delta_y=\liminf_{N\to\infty}\frac{|\mathcal{S}_y\cap[1,N]|}{N}, $$ then for Lebesgue almost all $y>1$, $$
\delta_y\ge \frac{1}{\log y+C \zeta(2)/\zeta(4)}, $$ where $C$ is the absolute Selberg-sieve constant defined in (2.2). The dependence on $y$ has the correct order as $y\to\infty$: a simple counting upper bound shows that no lower bound depending only on $y$ can have order larger than $1/\log y$. We also prove a complementary exceptional-base statement: even in the real-base setting one cannot expect density-one coverage in general. For the golden ratio $\varphi=(1+\sqrt5)/2$, the set of integers not representable as $$
p+\lfloor\varphi^k \rfloor,\qquad p\in\mathcal{P},
\quad k\ge1, $$ has positive lower density. The second problem is the square-free analogue of the power-of-two questions raised by Erdős in his 1950 paper on integers of the form $2^k+p$ and related problems. Erdős conjectured, in particular, that every sufficiently large odd integer should be a square-free integer plus a power of two; this fixed-base problem remains open. We prove a variable-base density-one analogue: there exists $a\in(2,3)$ such that $$
\#\{n\le x:n\notin \mathcal{Q}+\{\lfloor a^m\rfloor:m\ge1\}\}=o(x), $$ where $\mathcal{Q}$ denotes the positive square-free integers.