Periodic metallic structures are known to support resonant extraordinary transmission (EOT). When covered with graphene, these structures can be employed to effectively manipulate the light. In this work, we propose an analytical circuit model for graphene-covered one-dimensional metallic gratings. By using the circuit theory, we demonstrate that one-dimensional periodic array of cut-through slits which are covered by a continuous graphene sheet exhibit tunable EOT resonance whose amplitude, unlike its spectral position, can be dynamically tuned by varying the Fermi level of graphene. In this fashion, it is shown that placing a perfect reflector at the bottom of the graphene-covered metallic grating results in the realization of a graphene-based absorber. By utilizing the circuit theory, it is illustrated that perfect absorption in the structure is not exclusive to the TM polarization, but also the TE polarized plane waves can be completely absorbed by duly adjusting the Fermi level of graphene. Criteria for the enhanced absorption are accordingly presented. Results of this work may provide a useful tool for designing novel devices based on the graphene-covered metallic gratings such as filters, modulators and efficient absorbers.