In this paper we investigate the existence, uniqueness and approximation of solutions of delay differential equations (DDEs) with the right-hand side functions $f=f(t,x,z)$ that are Lipschitz continuous with respect to $x$ but only H\"older continuous with respect to $(t,z)$. We give a construction of the randomized two-stage Runge-Kutta scheme for DDEs and investigate its upper error bound in the $L^p(\Omega)$-norm for $p\in [2,+\infty)$. Finally, we report on results of numerical experiments.

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