In this article, we explore the Yukawa modification of the Casimir wormhole within the framework of f(R) gravity. To accomplish this, we adopt an approach proposed by Garattini (Eur Phys J C 81:824, 2021), wherein the original Casimir source is modified by a Yukawa term in three distinct scenarios. We utilize these Yukawa Casimir energy densities to assess the feasibility of traversable wormholes. The resulting shape function satisfies the criteria for traversability under the specified parameter values. Furthermore, we scrutinize the energy conditions and the violation of the null energy conditions suggests the presence of exotic matter within the Yukawa Casimir wormholes. Additionally, we analyze the embedding procedures for the wormhole geometry.

The ability of Big Bang Nucleosynthesis theory to accurately predict the primordial abundances of helium and deuterium, as well as the baryon content of the Universe, is considered one of the most significant achievements in modern physics. In the present study, we consider two highly motivated hybrid $f(T)$ models and constrain them using the observations from the Big Bang Nucleosynthesis era. In addition, using late-time observations of Cosmic Chronometers and Gamma-Ray Bursts, the ranges of the model parameters are confined which are in good agreement with early time bounds. Subsequently, the common ranges obtained from the analysis for early and late time are summarized. Further, we verify the intermediating epochs by investigating the profiles of cosmographic parameters using the model parameter values from the common range. From this study, we find the considered teleparallel models are viable candidates to explain the primordial-intermediating-present epochs.