We study the Dvoretzky covering problem for random covering sets driven by general Borel probability measures. As our main result, we solve the problem of covering analytic sets by random covering sets generated by arbitrary Borel probability measures on the real line. Prior to this work, a complete solution was not known for any singular measure. Our solution is potential theoretic and involves a generalisation of a notion of capacity in the work of Kahane, who solved the problem of covering compact sets in the classical setting where the random covering process is driven by the Lebesgue measure on the unit circle. One of our key innovations is a simple but powerful application of the Jankov-von Neumann uniformisation theorem, which we believe to have interest outside of this work. In addition, we determine the critical exponent for the covering problem for polynomially decreasing sequences $(cn^{-t})_n$ for random covering sets driven by Borel probability measures on $\mathbb{R}^d$. At exactly the critical exponent, the covering property generally depends on the constant $c>0$, and as an application of our main result, we determine the critical constant for random covering sets driven by natural measures on strongly separated self-conformal sets on the line. The critical constant depends on the multifractal structure of the average densities of the measure, and the result is new even for the simplest case of the Hausdorff measure on the Cantor set.