Let f be an entire function of finite order, let n≥1\documentclass[12pt]{minimal}
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\begin{document}$n\geq 1$\end{document}, m≥1\documentclass[12pt]{minimal}
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\begin{document}$m\geq 1$\end{document}, L(z,f)≢0\documentclass[12pt]{minimal}
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\begin{document}$L(z,f)\not \equiv 0$\end{document} be a linear difference polynomial of f with small meromorphic coefficients, and Pd(z,f)≢0\documentclass[12pt]{minimal}
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\begin{document}$P_{d}(z,f)\not \equiv 0$\end{document} be a difference polynomial in f of degree d≤n−1\documentclass[12pt]{minimal}
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\begin{document}$d\leq n-1$\end{document} with small meromorphic coefficients. We consider the growth and zeros of fn(z)Lm(z,f)+Pd(z,f)\documentclass[12pt]{minimal}
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\begin{document}$f^{n}(z)L^{m}(z,f)+P_{d}(z,f)$\end{document}. And some counterexamples are given to show that Theorem 3.1 proved by I. Laine (J. Math. Anal. Appl. 469:808–826, 2019) is not valid. In addition, we study meromorphic solutions to the difference equation of type fn(z)+Pd(z,f)=p1eα1z+p2eα2z\documentclass[12pt]{minimal}
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\begin{document}$f^{n}(z)+P_{d}(z,f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z}$\end{document}, where n≥2\documentclass[12pt]{minimal}
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\begin{document}$n\geq 2$\end{document}, Pd(z,f)≢0\documentclass[12pt]{minimal}
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\begin{document}$P_{d}(z,f)\not \equiv 0$\end{document} is a difference polynomial in f of degree d≤n−2\documentclass[12pt]{minimal}
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\begin{document}$d\leq n-2$\end{document} with small mromorphic coefficients, pi\documentclass[12pt]{minimal}
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\begin{document}$p_{i}$\end{document}, αi\documentclass[12pt]{minimal}
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\begin{document}$\alpha _{i}$\end{document} (i=1,2\documentclass[12pt]{minimal}
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\begin{document}$i=1,2$\end{document}) are nonzero constants such that α1≠α2\documentclass[12pt]{minimal}
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\begin{document}$\alpha _{1}\neq \alpha _{2}$\end{document}. Our results are improvements and complements of Laine 2019, Latreuch 2017, Liu and Mao 2018.
