On a Romanoff type problem of Erdős and Kalmár

Let $\mathbb{N}$ and $\mathcal{P}$ be the sets of natural numbers and primes, respectively. Motived by an old problem of Erd\H os and Kalmár, we prove that for almost all $y>1$ the lower asymptotic density of integers of the form $p+\lfloor y^k\rfloor (p\in \mathcal{P},k\in \mathbb{N})$ is positive.