If gadget failed to detonate, what was the plan, ? And was there a backup should one of the bombs dropped on Japan failed to detonate or airburst and instead just hit the ground without effect?
Richard Rhodes 'the making of the atomic bomb' suggests that the gun type bomb was held confidently to be so likely to work, given limited supplies of fissile material it made sense not to test it, compared to the sense of urgency to see an implosion device work, which is why the Trinity test was the implosion device, and why little boy was dropped first.
Basically, once the science was in on the purity of the uranium, the little boy gun-type bomb WAS the back up plan
https://www.amazon.com/Making-Atomic-Bomb-25th-Anniversary/dp/1451677618
[Edit: s/thin man/little boy/g the thin man was the aborted plutonium gun bomb, ditched because too reactive and would have been a fizzle]
Trinity/Gadget: The Gadget was tested because it was very uncertain. It was expected to not be very successful — the maximum likely expected yield was around 5,000 tons of TNT equivalent, with a possibility of it being as low as 400 tons of TNT, or a complete fizzle. If it had not worked, the US would have had only one atomic bomb (the gun-type design, which was expected to be about 15,000 tons of TNT equivalent). It is interesting in retrospect that the planning meetings in the run-up to Trinity assumed that gun-type bombs would be the "big" bombs, and that a lot more work would need to be done to get implosion weapons to that level. As for how that would have affected their policy towards Japan if the Gadget had failed, I do not know; presumably it would have, because they really did feel that their future policy forward hinged on knowing how successful the Gadget was, but they did not discuss it anywhere I have seen. The main difference would have been to their production line: to produce one Little Boy bomb required about 2 months of production time at full-speed. At full-speed, about 3 "Gadgets" could be produced per month. That's a big difference and it would have likely deprecated the US view of the importance of the atomic bomb in the end of the war, because the prevailing wisdom was that it would take multiple (perhaps as many as five to eight) nuclear weapons, paired with an invasion threat, to end the war.
Little Boy/Hiroshima. The uranium gun-type bomb was considered a sure thing. Unlike the Gadget, you could more or less determine whether it would work without any kind of extreme testing (you could test the components, like the gun system, in isolation; you could, using "dragon tickling" experiments, probe the limits of uranium criticality, etc.). To my knowledge they had no in-place plan for the event of a fizzle or non-detonation, but I suspect they would have immediately instigated a massive conventional bombing raid on Hiroshima, in an effort to make it difficult/impossible to recover the enriched uranium, which they recognized could be in turn used against them if recovered by the enemy.
Fat Man/Nagasaki. The risk of a fizzle or partial detonation was relatively high (one in ten or thereabouts), but unlike with the gun type bomb the chance of the Japanese being able to recover/use the material was low, since a partial detonation would scatter the plutonium widely (like a "dirty bomb") and the odds are that the 4 tons of TNT wrapped around it would detonate upon impact no matter what. Even if the plutonium could be recovered, there was no real fear the Japanese would be able to re-create the implosion mechanism necessary to use it in a small amount of time. Again, I have seen nothing to indicate they had hard contingency plans on the books, but I suspect they might have done a conventional raid afterwards just in case.
Long-story short: there isn't real documentation on their thoughts about #2 and #3 but they probably would have done something. #1 is interesting because it would have likely affected future planning.