The 2nd bomb was to try to make Japan believe we had an unlimited number of such bombs, & THEIR CITY OR TOWN COULD BE NEXT!
Mosta my life, I've heard from many a person, both former military & civilians alive during WW2 that, had Truman insisted that the proviso that Hirohito would be left on the throne under a shogun, that Japan woulda surrendered.
In late July, 1945, Japan had been overwhelmed everywhere but in China, & their leadership wanted to surrender. Their only interest was keeping the Emperor on the throne, & the whole nation woulda fought to the last bullet for that purpose. And it was truman's intent that Hirohito remain on the throne under a shogun, as Truman believed he would be of great help in reshaping Japan & suppressing the diehard militants. But he failed to insist on that proviso being included in the Potsdam Declaration, so Japan was determined to fight on. And even after the bombs, it took the Emperor himself to bring about the surrender.