This paper studies the cosmographic and matter bounce scenario in modified theory. The corresponding field equations are evaluated after considering special corrections of a Hubble parameter. The linear corrections to the Gauss-Bonnet gravity are being taken to analyze the behavior of Hubble and deceleration parameters. We derive dynamical parameters in a very general way to analyze different energy conditions that would lead to understanding the behavior of the equation of state parameters in cosmography. Finally, the removal of the initial singularity is observed to understand the late-time cosmic acceleration.