A corrected relativistic collision operator is used to derive a Fokker-Planck equation for the distribution function of relativistic suprathermal electrons in a weakly relativistic plasma, which is then solved by a procedure similar to that employed in Connor and Hastie [Nucl. Fusion 15 , 415 (1975)]. Analytical expressions are derived for the electron distribution function in plasmas with the electric field close to critical, which is typical of plasmas with grassy sawteeth on the Joint European Torus. A numerical solution is used for determining the normalization constant, which matches the relativistic region onto the weakly relativistic region. It is found that the scaling of the runaway rate with the electric field obtained by Connor and Hastie is a good approximation in spite of their use of an incomplete form of the collision operator not conserving number of particles. The present analysis determines the proportionality constant and introduces corrections to the earlier scaling of the runaway rate with respect to the electric field. The results obtained for the electron distribution function constitute a basis for studies of experimentally observed phenomena in near-threshold electric field plasmas with a significant suprathermal electron population.