  We study the dynamics of the planar circular restricted three-body problem in
the context of a pseudo-Newtonian approximation. By using the
Fodor-Hoenselaers-Perj\'es procedure, we perform an expansion in the mass
potential of a static massive spherical source up to the first non-Newtonian
term, giving place to a gravitational potential that includes first-order
general relativistic effects. With this result, we model a system composed by
two pseudo-Newtonian primaries describing circular orbits around their common
center of mass, and a test particle orbiting the system in the equatorial
plane. The dynamics of the new system of equations is studied in terms of the
Poincar\'e section method and the Lyapunov exponents, where the introduction of
a new parameter $\epsilon$, allows us to observe the transition from the
Newtonian to the pseudo-Newtonian regime. We show that when the Jacobian
constant is fixed, a chaotic orbit in the Newtonian regime can be either
chaotic or regular in the pseudo-Newtonian approach. As a general result, we
find that most of the pseudo-Newtonian configurations are less stable than
their Newtonian equivalent.
