What does an atom radiate: a wave packet or a single photon?
An excited atom radiates off a wave packet, that is, an oscillatory pattern of EM field which is effectively localized in space and time. It certainly does not emit the strictly monochromatic plane wave (which is infinite in space and time), nor a pointlike object (a pointlike distribution of a field includes Fourier components with arbitrary large energy). One can even speak of the "shape" of this wave packet: it typically has a sharp front and an exponentially decaying tail.
A separate question is what is the photon? I guess there is no universal answer; the actual thing that we call "the photon" depends on the situation.
When we quantize electromagnetic field, "the photon" means the excitation of the EM field with a fixed energy and momentum. (To be more precise, we introduce a regulator function that decays off at infinity, but then we consider everything in the limit when the regulator function tends towards identical unity. So, for all purposes we still can think of the monochromatic excitation). But when we discuss the normal modes of EM field with non-trivial boundary conditions, for example in a resonator inside a laser, the photons although being monochromatic are not spatially infinite plane waves anymore.
When we talk about emission and absorption of light quanta, which always proceeds in more or less localized wave packets, we usually call these wave packets the photons. In certain cases, for example in astrophysics, we can even speak of the time when a photon was emitted (although this is not a momentary act!).
All these subtle differences become even more delicate when discussing neutrino oscillation. There has been and still is a lot of confusion which originates from unjustified assumptions of how a neutrino is produced and detected (wave packets vs. plane waves, the moment and the position when the neutrino is emitted, etc.). Luckily, the EM field do not mix (at least in vacuum).