We consider a modification of classical branching random walk, where we add i.i.d. perturbations to the positions of the particles in each generation. In this model, which was introduced and studied by Bandyopadhyay and Ghosh (2023), perturbations take form $\frac 1 θ\log\frac X E$, where $θ$ is a positive parameter, $X$ has arbitrary distribution $μ$ and $E$ is exponential with parameter 1, independent of $X$. Working under finite mean assumption for $μ$, they proved almost sure convergence of the rightmost position to a constant limit, and identified the weak centered asymptotics when $θ$ does not exceed certain critical parameter $θ_0$. This paper complements their work by providing weak centered asymptotics for the case when $θ> θ_0$ and extending the results to $μ$ with regularly varying tails. We prove almost sure convergence of the rightmost position and identify the appropriate centering for the weak convergence, which is of form $αn + c \log n$, with constants $α$, $c$ depending on the ratio of $θ$ and $θ_0$. We describe the limiting distribution and provide explicitly the constants appearing in the centering.