On a Diophantine inequality involving a prime and an almost-prime

We prove that there are infinitely many solutions of $$\mathrm{\mid }{\lambda }_0+{\lambda }_{1}p+{\lambda }_2{P}_r\mathrm{\mid }<p^{-\tau },$$ where $r=3,$ $\tau =\frac{1}{118}$, and ${\lambda }_0$ is an arbitrary real number and ${\lambda }_1,{\lambda }_2\in \text{BR}$ with ${\lambda }_2\mathrm{\ne }0$ and $0>\frac{{\lambda }_1}{{\lambda }_2}$ not in $\mathbb{Q}$. This improves a result by Harman. Moreover, we show that one can require the prime $p$ to be of the form $\text{floor}{n}^c$ for some positive integer $n$, i.e. $p$ is a Piatetski-Shapiro prime, with $r=13$ and $\tau =\rho (c),$ a constant explicitly determined by $c$ supported in $(1,1+\frac{1}{149}]\mathrm{.}$