Some 30 years ago, an influential congressman named Jack Kemp gave a call to Bob Woodson, and in doing so became a Republican model for empowerment politics and minority outreach. A high-profile group of conservatives is staging a revival, looking to finally engage the modern left on the politics of the poor. Republican presidential candidates, pay attention.

As Hillary Clinton was rolling out her economic plan Monday—bashing Republicans as callous to the plight of struggling Americans—influential conservatives were taking...