Presence of Diabatic heating term in the continuity equation
In the book - Atmosphere-Ocean Dynamics, Volume 30  the author AE Gill defines the perturbation pressure in the paper that OP references as the following.

Consider a pressure of a reference system (ocean/atmosphere at rest) and let us call it 
    $$p_0$$.

It must be noted that a real atmosphere is never at rest and so we consider deviations from the reference system.

Since we are dealing with a vertical pressure gradient for OP's equation(2.5)(the gravitational force matches the vertical pressure gradient for hydrostatic large scale circulations) 

$$p_0$$ is a function of $z$ where z is the height coordinate. 

Then the perturbation pressure $p'$ is defined in the following way

$$p = p_0(z) + p' $$

So if you add the perturbation pressure to the reference pressure we get the deviation from the reference system.

Similarly a perturbation density can be defined in the following way

$$\rho = \rho_0(z) + \rho' $$

The basis for doing this comes from classical mechanics Perturbation Theory

The question is why is this being done ?

We are looking for solutions that are basically of a oscillatory nature. So in effect this gets rid of the non linear cross terms for which no oscillatory solution exists. 

In the specific case of 2.5 from the paper you referenced in addition to the Brunt–Väisälä frequency one needs to consider a "buoyancy forcing" especially in the tropics because of the large contributions from diabatic heating. We are essentially talking of an "open system" where in addition to adiabatic "work" being done you also have to account for heat exchange between a system and the ambient environment. In reality this is assumed as "pseudoadiabatic" process 

Then the question is how is the heating rate modeled ? In the AE Gill book this is given by the following equation 

$$ Q_H =  -L_V \frac{Dq_w}{Dt}$$ 

where $L_v$ is the latent heat of vaporization and $q_w$ is the saturation humidity where the word saturation means the saturation water vapor mixing ratios. 

Equation (2.5) from OP's paper is derived in Gill's book and is present in the 9th chapter(9.13.7) and will not be derived here but will be stated as is

$$\omega_n = \frac{\partial \widetilde \eta}{\partial t} + \widetilde b_n$$

Here $\widetilde \eta $ is the vertical coordinate(in OP's paper the vertical coordinate is pressure) and $b_n$ is the rate of change of buoyancy  per unit volume. This form is known as the buoyancy forced Shallow water equations

One can get more details about the perturbation theory for the atmosphere by looking at this link - Atmospheric Oscillations: Linear Perturbation Theory

Similarly if you want horizontal pressure perturbations you model the reference pressure as a function of x and y and then consider deviations thereof. 

Useful reading
What is the meaning of pressure in the Navier-Stokes equation?