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Suppose that $f(z)$ is a meromorphic function with hyper order $\sigma_{2}(f)<1$. Let $L(z,f)=b_1(z)f(z+c_1)+b_2(z)f(z+c_2)+\cdots+b_n(z)f(z+c_n)$ be a linear difference polynomial, where $b_1(z), b_2(z),\cdots, b_n(z)$ are nonzero small functions relative to $f(z)$, and $c_1, c_2,\cdots,c_n$ are distinct complex numbers. We investigate the uniqueness results about $f(z)$ and $L(z,f)$ sharing small functions. These results promote the { existing results} on differential cases and difference cases of Br\"{u}ck conjecture. Some sufficient conditions to show that $f(z)$ and $L(z,f)$ cannot share some small functions are also presented.