This paper investigates the zero distribution of the expression \( F = f^n P(z,f) - a \) , where \( f \) is a transcendental meromorphic function of hyper-order less than one, \( a \not \equiv 0 \) is a small function with respect to \( f \) , and \( P(z,f) \) is a non-vanishing delay-differential polynomial with small coefficients. We introduce the notions of sum-degree and sum-weight of \( P(z,f) \) , and use them to formulate conditions under which \( F \) has sufficiently many zeros. We also study paired delay-differential expressions of the form \( F_1 = f_1^{n_1} P_1(z, f_2) - a \) and \( F_2 = f_2^{n_2} P_2(z, f_1) - a \) , and establish conditions on \(P_1 (z,f_2)\) and \(P_2 (z,f_1)\) to ensure that at least one of the functions \( F_1 \) or \( F_2 \) has infinitely many zeros.
