First, I believe the OP is referring to a fourth atomic bomb. There were three constructed and used during the war, though one was a test rig.
- Trinity, an implosion plutonium bomb used for testing.
- Little Boy, a gun-type uranium bomb dropped on Hiroshima.
- Fat Man, an implosion plutonium bomb dropped on Nagasaki.
The gun-type uranium design was more conservative, but imploding a plutonium core was more efficient. Future bombs would all have plutonium cores.
Each bomb has a core of fissile material. If you're keeping track, that's two plutonium cores and one uranium core used in WWII. There were no more bombs completed during WWII, but the parts were including more plutonium cores. The most famous of these is the Demon Core.
Since this was the third plutonium core produced, and its fate was uncertain until very recently, I believe this might be the nugget of the claim of a "missing third bomb".
The Demon Core
Officially the Demon Core was HS-5,6;R-2 (plutonium-gallium hemispheres 5 & 6, plus a ring to separate them) and was being tested for use. HS-1,2 were used in Trinity, no ring. HS-3,4;R-1 were used in Fat Man. HS-7;R4 was being prepared (the records don't mention HS-8). So this was the third plutonium core produced.
It had originally been made in anticipation of needing to drop more bombs on Japan, but Truman put a halt to that, and shortly after Japan surrendered. Los Alamos had the parts of a third bomb ready to go, but it was never shipped during the war. They used the core for testing and it was involved in two major accidents that killed two scientists.
On August 21, 1945, Harry Daghlian was conducting experiments with the cores just below critical mass. He slipped and dropped a brick of tungsten carbide into the test assembly making the core briefly super-critical. He quickly pulled the brick away with his bare hand but had already received a lethal dose. He died 25 days later.
The super-critical event would make the core too radioactive for use for some time.
On May 21, 1946, Louis Slotin was again experimenting with the same core by hand. He had lowered the two halves of the core together separated by a screwdriver preventing them from going critical. He could adjust the separation by changing the angle of the screwdriver by hand, a procedure aptly known as "tickling the dragon's tail".
(A reconstruction of Slotin's fatal experiment. No, really! Bare hands and all.)
He slipped and the cores briefly went super-critical before he pulled them apart. Slotin received a lethal dose of radiation and died 9 days later.
The Third Core's Fate
The core was intended to be used at the third test, Charlie, of Operation Crossroads, but again time was needed for its radioactivity to decline after the criticality accident. Charlie test was scheduled for March 1st, 1947, but was cancelled after Baker (the second test) was unexpectedly radioactive and the test site could not be cleaned up in time.
It was not known what became of the Demon Core, the early US nuclear program was very sketchy. Finally in 2016 interviews and searches of the archives revealed the Demon Core was ultimately melted down and reused for another core.