We consider the problem of designing incentive-compatible, ex-post individually rational (IR) mechanisms for covering problems in the Bayesian setting, where players' types are drawn from an underlying distribution and may be correlated, and the goal is to minimize the expected total payment made by the mechanism. We formulate a notion of incentive compatibility (IC) that we call {\em support-based IC} that is substantially more robust than Bayesian IC, and develop black-box reductions from support-based-IC mechanism design to algorithm design. For single-dimensional settings, this black-box reduction applies even when we only have an LP-relative {\em approximation algorithm} for the algorithmic problem. Thus, we obtain near-optimal mechanisms for various covering settings including single-dimensional covering problems, multi-item procurement auctions, and multidimensional facility location.