Let f be an entire function of finite order, let ngeq 1, mgeq 1, L(z,f)not equiv 0 be a linear difference polynomial of f with small meromorphic coefficients, and P_{d}(z,f)not equiv 0 be a difference polynomial in f of degree dleq n-1 with small meromorphic coefficients. We consider the growth and zeros of f^{n}(z)L^{m}(z,f)+P_{d}(z,f). 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 f^{n}(z)+P_{d}(z,f)=p_{1}e^{alpha _{1}z}+p_{2}e^{alpha _{2}z}, where ngeq 2, P_{d}(z,f)not equiv 0 is a difference polynomial in f of degree dleq n-2 with small mromorphic coefficients, p_{i}, alpha _{i} (i=1,2) are nonzero constants such that alpha _{1}neq alpha _{2}. Our results are improvements and complements of Laine 2019, Latreuch 2017, Liu and Mao 2018.