  This paper contains an error analysis of two randomized explicit Runge-Kutta
schemes for ordinary differential equations (ODEs) with time-irregular
coefficient functions. In particular, the methods are applicable to ODEs of
Carath\'eodory type, whose coefficient functions are only integrable with
respect to the time variable but are not assumed to be continuous. A further
field of application are ODEs with coefficient functions that contain weak
singularities with respect to the time variable. The main result consists of
precise bounds for the discretization error with respect to the
$L^p(\Omega;\mathbb{R}^d)$-norm. In addition, convergence rates are also
derived in the almost sure sense. An important ingredient in the analysis are
corresponding error bounds for the randomized Riemann sum quadrature rule. The
theoretical results are illustrated through a few numerical experiments.
