  Let $(u_n)_{n \ge 0}$ be a nondegenerate Lucas sequence and $g_u(n)$ be the
arithmetic function defined by $\gcd(n, u_n).$ Recent studies have investigated
the distributional characteristics of $g_u$. Numerous results have been proven
based on the two extreme values $1$ and $n$ of $g_{u}(n)$. Sanna investigated
the average behaviour of $g_{u}$ and found asymptotic formulas for the moments
of $\log g_{u}$. In a related direction, Jha and Sanna investigated properties
of $g_{u}$ at shifted primes. In light of these results, we prove that for each
positive integer $\lambda,$ we have $$\sum_{\substack{p\le x\\p\text{ prime}}}
(\log g_{u}(p-1))^{\lambda} \sim P_{u,\lambda}\pi(x),$$ where $P_{u, \lambda}$
is a constant depending on $u$ and $\lambda$ which is expressible as an
infinite series. Additionally, we provide estimates for $P_{u,\lambda}$ and
$M_{u,\lambda},$ where $M_{u, \lambda}$ is the constant for an analogous sum
obtained by Sanna [J. Number Theory 191 (2018), 305-315]. As an application of
our results, we prove upper bounds on the count $\#\{p\le x : g_{u}(p-1)>y\}$
and also establish the existence of infinitely many runs of $m$ consecutive
primes $p$ in bounded intervals such that $g_{u}(p-1)>y$ based on a
breakthrough of Zhang, Maynard, Tao, et al. on small gaps between primes.
Exploring further in this direction, it turns out that for Lucas sequences with
nonunit discriminant, we have $\max\{g_{u}(n) : n \le x\} \gg x$. As an
analogue, we obtain that that $\max\{g_u(p-1) : p \le x\} \gg x^{0.4736}$
unconditionally, while $\max\{g_u(p-1): p \le x\} \gg x^{1 - o(1)}$ under the
hypothesis of Montgomery's or Chowla's conjecture.
