When an airplane is disturbed from a longitudinal equilibrium position, what sort of motion will the aircraft experience? Will it be a phugoid or a short period motion? Or will it be both?
The separation into short-period and phugoid motion is somewhat artificial. There is just motion. It just happens that when you fly a 'real' airplane (or solve its equations of motions), you'll have quite different timescales for angular motion (isolated pitch; low seconds) and linear motion (changes of altitude and speed; tens of seconds and even minutes).
Broadly speaking, heavier and faster airplanes with swept wings demonstrate more of this separation. At the other end of the spectrum, smaller airplane models have similar timescales for both the short- and long-period motion, and for them the separation is not meaningful (which makes it more difficult for analysis!)
So, to answer the question, both will usually occur. In some cases, depending on the airplane and your point of interest, you can ignore one or the other. On airplanes where separation is significant and motion is stable and well damped, it may also be possible to produce a disturbance that results in (practically) one kind of motion: either short-period (e.g. a small elevator impulse) or long-period (e.g. a brief application of pitch-neutral air brakes).
Wikipedia suggests that both happen, and that short period motion should be damped out in a second or so for an aircraft to be certified, and that phugoid motion should be gentle enough that a pilot corrects it almost without thinking.
This is the result of careful design and testing, plenty of prototypes weren't that forgiving.
The answer depends on whether the plane is designed for quick responsiveness to control inputs or hands-off stability. Hands-off stability requires the plane to exhibit critical damping in its responses to perturbations whereas quick responsiveness requires something closer to underdamped response, in which control inputs from the pilot are required to prevent oscillation or divergence.