In [Ann. Math. (2) 70, 9–42 (1959;

Zbl 0088.28505)]

*W. K. Hayman* proved (among other results) that a differential polynomial

${f}^{n}+a{f}^{\text{'}}-b$ with constant coefficients

$a,b$ admits infinitely many zeros, provided that

$f$ is transcendental entire and

$n\ge 3$ (or

$n\ge 2$ if

$b=0$). The authors consider the difference counterpart of the expression above:

${f}^{n}\left(z\right)+f(z+c)-f\left(z\right)-b,n\ge 3$ (or

$n\ge 2$ if

$b=0$). They prove that it has infinitely many zeros, provided that

$f$ is a transcendental entire function of finite order, not of period

$c$. It is shown that one can replace

$b$ in this result with a nonzero function

$b\left(z\right)$, small compared to

$f$. The authors also prove a result related to what can be called a difference counterpart of the

*R. Brück* conjecture, see [Result. Math. 30, No.1–2, 21–24 (1996;

Zbl 0861.30032)].