I've never heard of either bomb malfunctioning in any way. The closest I can think of is that the second (Nagasaki) bomb was dropped slightly off-target, and thus may not have been as overwhelmingly destructive as intended. But they both exploded just fine.

Both of the bombs worked as expected. The Little Boy bomb was a very inefficient bomb design (in that very little of its bomb fuel reacted), but was nearly on-the-nose in terms of its expected explosive output (about 15 kilotons of TNT equivalent). The Fat Man bomb also worked as expected (a more efficient design, with an output of about 20 kilotons of TNT from about 10X less bomb fuel), but it was not placed accurately, and the geography was not ideal at Nagasaki (Hiroshima was a large "bowl" of a city, whereas Nagasaki had valleys that constrained the blast damage). The result is that the total area destroyed by the bomb was smaller than at Hiroshima (by a considerable amount), but the destruction there was much more intense (the hills reflected the blast damage inward).

There were many mishaps on the Nagasaki mission, but the bomb functioned as designed.