This paper introduces a randomized tamed Euler scheme tailored for Lévy-driven stochastic differential equations (SDEs) with superlinear random coefficients and Carathéodory-type drift. Under assumptions that allow for time-irregular drifts while ensuring appropriate time-regularity of the diffusion and jump coefficients, the proposed scheme is shown to achieve the optimal strong $\mathcal{L}^2$-convergence rate, arbitrarily close to $0.5$. A crucial component of our methodology is the incorporation of drift randomization, which overcomes challenges due to low time-regularity, along with a taming technique to handle the superlinear state dependence. Our analysis moreover covers settings where the coefficients are random, providing for instance strong convergence of randomized tamed Euler schemes for Lévy-driven stochastic delay differential equations (SDDEs) with Markovian switching. To our knowledge, this is the first {work} that addresses the case of superlinear coefficients in the numerical analysis of Carathéodory-type SDEs and even for ordinary differential equations.