You cannot just add the ideal gas pressure and degeneracy pressure together for any particular species of particle. (Well, you can, but it would be a poor approximation to the actual pressure).
The trouble is that they are not separate things. In general, the electrons in the gas are partially degenerate and always obey Fermi-Dirac statistics. Unfortunately there is no analytic solution that gives you pressure as a function of temperature and density; you either have to use Tables of pre-calculated results, or there are analytic approximations that work well in certain regimes.
Clayton's Principles of Stellar Evolution and Nucleosynthesis contains some of these tables and also some better analytical approximations that work in certain regimes of (partial) degeneracy.
What is true is that you are fine to add to this pressure, the radiation pressure due to photons and the ideal gas pressure of the ions.
EDIT: I realise that what may be troubling you is not the above, but why you can add the separate pressures together at all (eqn 122 in the notes you reference). The reason can be traced to eqn 31 in the same notes. This says that the pressure is an integral over the number density and momenta of individual particles. Note that this equation does not contain anything that identifies the particles (i.e. their mass). The integral can therefore be treated as a summation over all particles, which in turn can be separated into three integrals that cover the ions, electrons and photons separately.
