  In this paper, we investigate the uniqueness problem of difference
polynomials $f^{n}(z)P(f(z))L_c(f)$ and $g^{n}(z)P(g(z))L_c(g)$, where
$L_c(f)=f(z+c)+c_0f(z)$, $P(z)$ is a polynomial with constant coefficients of
degree $m$ sharing a small function with the notions of weakly weighted sharing
and relaxed weighted sharing and obtained the corresponding results, which
improve and extend some recent results due to Sahoo and Biswas (Tamkang J.
Math., \textbf{49}(2), 85--97 (2018)).
