The nonlinear propagation of electrostatic solitary waves is studied in a collisionless electron-positron pair plasma consisting of adiabatic cool electrons, mobile cool positrons (or electron holes), hot suprathermal electrons described by a κ distribution, and stationary ions. The linear dispersion relation derived for electrostatic waves demonstrates a weak dependence of the phase speed on physical conditions of positrons in appropriate ranges of parameters. The Sagdeev's pseudopotential approach is used to obtain the existence of electrostatic solitary wave structures, focusing on how their characteristics depend on the physical conditions of positrons and suprathermal electrons. Both negative and positive polarity electrostatic solitary waves are found to exist in different ranges of Mach numbers. As the positrons constitute a small fraction of the total number density, they slightly affect the existence domains. However, the positrons can significantly change the wave potential at a fixed soliton speed. The results indicate that the positive potential can largely be grown by increasing the electron suprathermality (lower κ) at a fixed true Mach number. It is found that a fraction of positrons maintain the generation of positive polarity electrostatic solitary waves in the presence of suprathermal electrons in pair plasmas.