Denote by
$\mathbb{P}$ the set of all prime numbers and by
$P(n)$ the largest prime factor of positive integer
$n\geq 1$ with the convention
$P(1)=1$. In this paper, we prove that, for each
$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$, there is a constant
$c(\unicode[STIX]{x1D702})>1$ such that, for every fixed nonzero integer
$a\in \mathbb{Z}^{\ast }$, the set
$$\begin{eqnarray}\{p\in \mathbb{P}:p=P(q-a)\text{ for some prime }q\text{ with }p^{\unicode[STIX]{x1D702}}<q\leq c(\unicode[STIX]{x1D702})p^{\unicode[STIX]{x1D702}}\}\end{eqnarray}$$ has relative asymptotic density one in
$\mathbb{P}$. This improves a similar result due to Banks and Shparlinski [‘On values taken by the largest prime factor of shifted primes’, J. Aust. Math. Soc.82 (2015), 133–147], Theorem 1.1, which requires
$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.0606\cdots \,)$ in place of
$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$.