  We prove that there are infinitely many solutions of $$
|\lambda_0+\lambda_1p+\lambda_2P_r|<p^{-\tau}, $$ where $r=3,$
$\tau=\frac1{118}$, and $\lambda_0$ is an arbitrary real number and
$\lambda_1,\lambda_2\in\BR$ with $\lambda_2\neq0$ 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
$\floor{n^c}$ 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 $\left(1, 1+\frac1{149}\right].$
